Tcl, 39 bytes

proc H n {expr $n?$n**$n*\[H ($n-1)]:1}

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Periodically, 52 bytes

Ti(HICa)3HHIBrOBO[B[BONI]2OOBIICaIBICINOFOOOKI]0IICl

explanation soon

Pip, 9 bytes

Y\,a$*yEy

Takes input via command-line argument. Attempt This Online!

Explanation

 \,a       Inclusive range from 1 to command-line arg
Y          Yank that into y
      yEy  y to the power of y
    $*     Fold on multiplication

☾, 7 chars (22 bytes)

󷺹󷸺ᴍ⌃꜠⨀¹

Test it here.

For some reason, you have to specify an initial value for the product when the length of the input is 0, even though it should be 1 by default.

Bespoke, 168 bytes

when I speak,I am MORE excited
power of self is more exciting
how it swells in strength,capacity&magnitude
it brings everyone measures so much greater
but do I exceed u

A slight modification of my factorial answer.

CASIO BASIC (CASIO fx-9750GIII), 20 bytes

?→N
Prod Seq(K^K,K,1,N,1

AWK, 28 bytes

{for(x=1;i++<$1;)x*=i^i}$0=x

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{for(x=1;  # can't multiply by zero
i++<$1;)   # increment to input
x*=i^i}    # hyperfactorial
$0=x       # set output

sed 4.2.2 -r, ^{_{291 287 276 260 255 251 248 240 233 228 226 220 219 212 209 208 207 196 193 191 184 181 177 175 173 170 169 165 164}} 163 bytes

h;G;G
:;s/.//;x;s/\s.*//
:a;s/#*/& &/;T;x;s/^##/#/;x;ta
x;s/#.#//;H;g;s/ //g;/\n##+\n/t;T
s/(#+)\n#.#(#*)/\2\n\2\n\2	\1/;h;t
s/.#?\n+/#/;h;T;s/.*	#//;x;s/	#*$//;ta

Try it online! (cases 0-4; slow)
Try it online! (cases 0-5; fast, +1 byte)
Try it online! (cases 0-7; calculate the powers but don't multiply them together)

takes input and output as unary #s. TIO links include conversion to/from decimal in header/footer.

after two months and 100 bytes of golfing, I think it's high time to actually write out an explanation. normally, I add the explanation in comments next to the code, but I think explaining the data format first will work better with this one. the PATTERN and HOLD spaces are switched around a lot, but let's call this the initial PATTERN space:

add
mult
pow	prev

each 'variable' is a unary number of #s which keeps track of how many more times we need to preform each operation. the first one used is add, where the a loop adds the HOLD space (which is currently pow) to itself and decrements add. once add is 0 (ie HOLD now equals add * pow), mult gets decremented by one and add is set to HOLD. once mult is 1 (ie HOLD now equals pow ^ pow), pow is decremented, add and mult are set to pow, and prev gets set to HOLD\tprev (where \t is a tab). once pow is 0, we have calculated n^n for every n, starting at the input and going all the way down to 0, with the results separated by tabs. finally, the last time we loop through is to multiply these results together, which we do by first removing all newlines to let it pass through the first checks, and loading the final result into the HOLD space while deleting it from the PATTERN space until there is only one result left.

description of code by line:

1: sets the format up like how I described as 'initial PATTERN space' where the HOLD space is the input and the PATTERN space is the input repeated three times
2: a 'prefix' which takes care of an off-by-one error and discards junk that accumulates in the HOLD space
3: the `add` loop, which ends with the HOLD space equaling HOLD space * first line of PATTERN space
4: the `mult` loop, which ends with the HOLD equaling HOLD space ^ second line of PATTERN space
5: the `pow` loop, which saves the result of the `mult` loop after a tab and restarts everything with a decremented input
6: the `final` loop, which removes all the newlines to bypass the `mult` and `pow` loops, and multiplies the first result to the last result until there is only one result

Rust, 55 bytes

let x=|n|(1..=n).map(|x:u32|x.pow(x)).product::<u32>();

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Swift 6, 69 bytes

import Foundation
let h={(1...$0).map{pow(.init($0),$0)}.reduce(1,*)}

ARBLE, 18 bytes

range(1,n)/(y*z^z)

Simply reduces the range of numbers [1,n] via the function (y,z)=>y*pow(z,z)

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Clojure, 45 bytes

#(apply *(for[i(range 1(inc %))_(range i)]i))

TIO. With one extra byte (use *') you get arbitrary precision.

Setanta, 48 bytes

gniomh f(n){toradh(n&cmhcht@mata(n,n)*f(n-1))|1}

Try on try-setanta.ie

Vyxal 3, 4 bytes

ɾ:*Π

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Uiua, 12 bytes, 7 characters

/×ⁿ.+1⇡

Try it on Uiua Pad!

Explained

Uiua reads code from right to left, so this code will also be explained in that order.

+1⇡ // Make an array of numbers from 0 to n and increment each element 
.   // Create a copy of the array
ⁿ   // Raise each element to the power of itself
/×  // Reducing function: multiplicate each element by the next element on the array

Kotlin, 59 58 bytes

fun h(i:Double):Double=if(i<1)1e0else Math.pow(i,i)*h(i-1)

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Longest submission yet, but thought I might as well give Kotlin a shot. Used the = format for functions to eliminate 2 brackets as well as the return.

UPDATE: it's 2024 and I've been golfing the same answer for three and a half years. Along comes RubenVerg and absolutely destroys my efforts within a minute with -1 byte. Turns out it's a new feature from Kotlin 2.0 (which was released a month or so ago), which is why I haven't gotten it earlier.

C#, 77 bytes

n=>Enumerable.Range(0,n+1).Select(i=>Math.Pow(i,i)).Aggregate(1.0,(a,b)=>a*b)

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Boring answer, but C# doesn't really have fancy ways of doing this. Even with the addition of the range operator (0..n+1), that produces a System.Range, which is not an IEnumerable and so can't be used as an iterator :(

1.0 in the call to Aggregate because it expects the start value and the return type of the predicate to be exactly equal, and Math.Pow returns doubles, not ints.

K (ngn/k), 6 bytes

*/&!1+

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• !1+ generate 0..x (e.g., with an input of 4, generate 0 1 2 3 4)
• & take n copies of each n, e.g. zero copies of 0, one copy of 1, two copies of 2, etc.)
• */ calculate (and implicitly return) the product

Arturo, 29 22 19 bytes

$=>[∏map&'x->x^x]

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Japt, 6 5 bytes

ô*² ×

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awk

a hyperfactorial function in POSIX-compliant awk syntax that bypasses any and all usage of the exponentiation operator, without sacrificing efficiency :

jot 20 | gawk -Mbe '

function _____(__, _, ___, ____) {

    if ((_ = ___ = __ = int(__)) <= (____ += ____ = !!_))

        return (_ = __ < ____) ? _ : __ + __
 
    while (____ < --_)___ *= (__ *= _) * (__ *= --_)

    return _ < ____ ? ___ *  __ \
                    : ___ * (__ *= _) * __

} $++NF = _____($!_)'

1 1
2 4
3 108
4 27648
5 86400000
6 4031078400000
7 3319766398771200000
8 55696437941726556979200000
9 21577941222941856209168026828800000
10 215779412229418562091680268288000000000000000
11 61564384586635053951550731889313964883968000000000000000
12 548914237009501581804104224704637116078267727827959808000000000000000
13 166252458044258018207674078620690924617735088270974773221032328167424000000000000000
14 1847398448553592782012673311296877599223436283900539192451554723195762806303473270784000000000000000

UPDATE

adding a functional programming style recursive function in awk (but this version using the exponentiation operator) :

func __(_) { return _^_--*(1<_?__(_):1) }

awk's syntax looks really modern for a 50-year old language

Rust, 40 bytes

|n|(1u32..=n).map(|n|n.pow(n)).product()

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Staightforward functional approach. Follows the rules for type inference. As the program takes and returns a u32, only calls up to H(5) are supported. This is because {integer}::pow takes a u32, and casting would cost five bytes.

BLC, 8 bytes (61 bit)

0000010101110000001011010011100000010111101100111010001100010

Works for Church encoded numerals with 1=i. The code is derived from the common fac function. Degolfed to bruijn:

fac [[1 [[id (1 [[2 1 (1 0)]])]] [1] [0]]]
    id 0

hyperfac [[1 [[exp (1 [[2 1 (1 0)]])]] [1] [0]]]
    exp (0 0)

Uiua, 7 bytes

/×ⁿ.+1⇡

Try it!

/×ⁿ.+1⇡
    +1⇡  # 1..n
   .     # duplicate
  ⁿ      # power
/×       # product

Thunno 2, 4 bytes

RD*p

Explanation

RD*p  # Implicit input            6
R     # Push the one-range        [1, 2, 3, 4, 5, 6]
  *   # Each to the power of...
 D    # ...itself                 [1, 4, 27, 256, 3125, 46656]
   p  # Product of the list       4031078400000
      # Implicit output

Lua, 38 bytes

a=1 for i=1,...do a=a*i^i end print(a)

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SageMath, 35 bytes

Without using NATIVE python module or syntax

Golfed vesrion. Run it on SageMathCell!

f=lambda n:prod(i^i for i in[1..n])

Pyt, 3 bytes

řṖΠ

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ř            implicit input (n); push [1,2,3,...,n]
 Ṗ           raise each element to the power of itself
  Π          take the product of all of them; implicit print

><>, 42 bytes

1$01.11-1<
>:>:?v~:?^~l1=?n*a1.
1-^  >$:}$

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Explanation

Initializes product as 1 to handle the n=0 special case.
Loops over x from n down to 1 and puts x copies of x on the stack.
Ex: n=3 results in 3,3,3,2,2,1
Then takes the product of all numbers on the stack to get the result.

K (ngn/k), 15 bytes

+/*/'{x#x}'1+!:

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Another quick answer.

Explanations:

+/*/'{x#x}'1+!:  Main function. Takes implicit input with :
             !   Range from [0..input-1]
           1+    + 1 to turn it into [1..input]
          '      For each number...
     {   }       Execute a function that...
        x        Takes the number as implicit variable x
      x#         And duplicate them x amount of times
    '            For each of the lists inside...
  */             Fold and multiply them
+/               Sum

J, 11 bytes

[:*/1^~@+i.

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[:*/1^~@+i.
[:          NB. enforce f (g x)
  */        NB. product reduce
         i. NB. range 0..x-1
    1   +   NB. add one
       @    NB. then
     ^~     NB. raise each item to itself

Cognate, 36 bytes

(Let N;For Range 1 + 1 N(* ^ Twin)1)

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Knight (v2.0-alpha), 27 bytes

;;=i^=n+0PnW=n-nT=i*i^n nOi

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Knight, 30 bytes

;=p=w 1;=qP;W<w q=p*p^=w+wTwOp

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Fig, \$4\log_{256}(96)\approx\$ 3.292 bytes

r^ua

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Beats osabie and Vyxal, still longer than Jelly tho.

r^ua
   a # Range [1, n]
 ^u  # Exponentiate by self
r    # Product

Nibbles, 2 bytes

$*@^

Note that this problem influenced how nibbles decides to use implicit args from implicit ops. I'm currently thinking of getting rid of implicit fold since it is uncommonly used though.

     # implicit fold since accumulator is used
     # implicit range from 1 since there's things after an integer 
$    # first input integer
 *   # mult
  @  # accumulator from fold
   ^ # exponentiation
     # implicit $ since need another arg to ^
     # implicit $ since need another arg to ^

Desmos, 19 bytes

f(k)=∏_{n=1}^kn^n

Try It On Desmos!

Try It On Desmos! - Prettified

Swift 2.2, 57 bytes

In an unexpected twist, I'm putting the old version of Swift nobody uses anymore as my main answer.

func h(x:Double)->Double{return x==0 ? 1:pow(x,x)*h(x-1)}

Swift >=3.0, 77 bytes

In Swift 3, Foundation/Foundation.h is no longer exposed by default, and you have to explicitly import Foundation:

import Foundation
func h(_ x:Double)->Double{return x==0 ? 1:pow(x,x)*h(x-1)}

Swift >=5.1, 70 bytes

In Swift 5.1, the return keyword becomes optional:

import Foundation
func h(_ x:Double)->Double{x==0 ? 1:pow(x,x)*h(x-1)}

TIO doesn't support all these Swift versions, but I found this website which does: SwiftFiddle.

PHP 5.6 (50 chars)

Given $argv[1] as a command line argument :

$r=($n=$argv[1])**$n;for(;$n--;)$r*=$n**$n;echo$r;

PHP 5.6 (42 chars)

To get the answer as a variable, the function a($input) return the answer recursively :

function a($n){return$n?$n**$n*a($n-1):1;}

PHP 7 (38 chars)

The function $_ENV is like it's PHP5.6 equivalent function.

$_ENV=fn($n)=>$n?$n**$n*$_ENV($n-1):1;

Note : The $_ENV overwriting is a bad practice.

Python 3.8 (pre-release), 34 33 bytes

f=lambda n:not(n+1)or n**n*f(n-1)

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Brachylog, 7 bytes

⟦gjz^ᵐ×

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Explanation

⟦            Range: [0, …, Input]
 g           Nest in a list: [[0, …, Input]]
  j          Juxtapose: [[0, …, Input], [0, …, Input]]
   z         Zip: [[0, 0], …, [Input, Input]]
    ^ᵐ       Map power: [1, …, Input^Input]
      ×      Multiply: 1 × … × Input^Input

Python + mpmath, 28 bytes

from mpmath import*
hyperfac

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Factor + math.factorials, 15 bytes

hyper-factorial

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Built-in.

Factor + math.unicode, ³³ 27 bytes

[ [1,b] [ dup ^n ] map Π ]

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This could have been 26 bytes, but ^ returns NaN for 0^0. Factor is the only language I know of that doesn't just return 1 for that. So I had to use ^n instead, which is a helper word used to implement ^ that hasn't been 'fixed' yet.

• [1,b] A range from 1 to whatever is on top of the data stack, inclusive.
• dup ^n Raise an integer to itself.
• [ ... ] map Apply a quotation to each element of a sequence, collecting the results in a new sequence of the same length.
• Π Product of a sequence.

Vyxal, 4 bytes

ƛe;Π

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Excel, 33 bytes

=LET(x,SEQUENCE(A1),PRODUCT(x^x))

Link to Spreadsheet

BQN, 10 bytes SBCS

{×´⋆˜1+↕𝕩}

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Explanation

{×´⋆˜1+↕𝕩}

{        }   # Anonymous function / block
       ↕𝕩    # 0..𝕩
     1+      # To 1...𝕩
   ⋆˜        # Raise to the power over itself (1*1,2*2,3*3...𝕩*𝕩)
 ×´          # Fold using ×
```

tinylisp, 51 bytes

(load library
(d H(q((N)(i N(*(pow N N)(H(dec N)))1

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Ungolfed

(load library)            ; for *, pow, and dec

(def hyper                ; name something
  (lambda (n)             ; a lambda function that takes one parameter n
    (if n                 ; if n...   
      (*                  ; ...is non-zero, multiply...
        (pow n n)         ; ...n raised to itself...
        (hyper (dec n)))  ; by the hyperfactorial of n-1
      1)))                ; otherwise if n is zero, return 1

Vyxal, 7 bytes

?ɾƛ:e;Π

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Will add an explanation soon.

Scala, 40 38 bytes (supporting Big Integers)

1.to(_).map(x=>BigInt(x)pow x).product

Provides exact results up to crazy large numbers with several billions of digits, namely up to Hyperfactorial(2^31-1).

Saved 2 bytes thanks to @user.

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Red, 40 bytes

func[n][a: 1 repeat i n[a: i ** i * a]a]

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J-uby, 26 bytes

:+&1|:*|:*&->r{r**r}|:/&:*

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I want to remove the inner lambda, but ~:** doesn't seem to work in its place. Might be a bug or I'm missing some argument error.

PHP -F, 42 39 bytes

for($r=1;$argn-$n++;)$r*=$n**$n;echo$r;

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No PHP yet, well this is The Answer! Unfortunately I had to resort to incrementing first, because of a weird operator precedence when using $n**$n-- ($n should decrement after the exponentiation if I RTM correctly, but actually it decrements first)

EDIT: nice improvement from Hydrazer, -3 bytes with a way to avoid using $n twice in the loop declaration

Rust, 57 45 bytes

@alephalpha gr8 m8 8 outta 8 (they use fold/reduce with a clojure closure instead of recursion to avoid function boilerplate)

|n:i128|(1..n).fold(1,|p,i|i.pow(i as u32)*p)

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oeuf

Python 3, 47 41 bytes

new code

z=lambda k,a=1: z(k-1,a*k**k) if k else a

old code (for python 3.8)

lambda k,a=1:[a:=a*i**i for i in range(k+1)][k]

Try it here!

Explanation: Looping through and getting each element, we assign and insert with the walrus operator (I love this thing haha), and return the last element (Which, here, is k saving one more byte instead of using -1!). I am also making use of the default value for the lambda to assign 1 to a, since I can't assign it inside the lambda. Note that i lose 2 bytes in range(k+1) since the range doesnt go to the last number, but I don't see a way around it.

Getting rid of both the walrus (rip) and the range (yes!), it is done with a recursive lambda, saving a whole 6 bytes! We multiply the values and store the result, which we send to the next iteration of the code until k = 1, when we can just ignore and return a since 1^1*a = a.

dc, 24 bytes

?[1+[d1-d1<xr1-d^*]dsxxp

Explanation

This is a complete program, that reads input from stdin and writes the result to stdout.

?                       # input
1+                      # avoid H(-1)
[                       # function x
  d1-                   # 
  d1<x                  # recurse       : n n-1 ... i+1 H(i-1)
  r1-                   #               : n n-1 ... H(i-1) i
  d^*                   #               : n n-1 ... H(i)
]dsxx                   # define and execute
p                       # output

Output

H(n) for n = 0..12

0
1
4
108
27648
86400000
4031078400000
3319766398771200000
55696437941726556979200000
21577941222941856209168026828800000
215779412229418562091680268288000000000000000
61564384586635053951550731889313964883968000000000000000
548914237009501581804104224704637116078267727827959808000000000000000

My machine takes almost 0.7 seconds to compute H(200), but it does produce the correct result:

114256002424540571298977792354054343003217988447353197803701128087779\
727310773544883222069862169886712875089569689613436517195625058699339\
831616247967378727547253765461789388643475674400611735920545310021028\
378006078740483920058608872101517118552137511462765664553099379243826\
960171887170750794198373595289872526218671094534622811250467573078955\
097722440499909299236803909456886903559422482755722874996888761181995\
575299415208051459908113833797191408981585936952677202993799475038316\
559629150663378834397372775965856682894873495939410259095804412984759\
376802645433107086846121910224544820403394942396165166503784543183923\
373316267239065885057694972898540625272750046246522113004186914354002\
967536413371473458112220769020094756837578527902012213205896270144393\
863599279497755781359466901673079326909202551645285985747551170700716\
842425698975536766498808553756048408306339794216494138288030784286578\
757511579359563917784149939178533514638276262178221710320662541522716\
166268310550789650887610134719729891484651131282509122835031254510412\
859215509732637002095449585308892098589802359045914197581197309643963\
042570582527203047381981490652454323893038445041694758990588984561508\
024977520153869302479414855203541538559245994267482585355022642863859\
784222422005021616964777696793772349735028255170680455712484791660880\
381133647422166704047722699070109094074510187247343055332257496929374\
982064491045838594959100552218107373079642910265887514107042314387137\
231583168255471546888411138370063211060324585902513206145493958317539\
840648935101056767774783896580016743810934173275647035655085909723344\
400585286826414068777563965071196389690941135698863589037499407785370\
931798703443963281315725441703395909623807818562899480337755459429222\
263976154559801703132418440141220192063353095047779835936440544198953\
496106173470542538936478118001519398934406367850808122800102137273226\
951723289344106737093183594520201847490204826536442751006791251960765\
409736570349501810834880546030444722460505622335562771218876927864732\
533831989277488466279926764165356648286028875357501991394907871736473\
399337480995325006099531629623628775557151039011468601336589825839738\
631718648726576082486937471392562052316112813773965598704019732181766\
979636498126699138316658703065343361486555049963171513229287190102366\
294038070909517003192684859103788524924734600167422726737999487560861\
040910125441478016464932490390352767808126453168977619717961242515337\
678589841167345358713873500430963639031923908700261694508782150681050\
006111529466925629252138641356979249757548377531461415227595617743646\
019706623647306680452085764528828628378729906162793797766057848324098\
880977726680150730390074716141211796155230858668012021940942979991319\
892197094655756666501455192326506825509530719155451676296396102560013\
841584265351485313695020472184192704099107932797640710173835532960606\
518538605516574602321430208034314359982648827905481627199797808702727\
344458806167205608568625797981988480548489292548067131307825217362115\
487277151573038609637079476077578954700658210221211043513144482137132\
113794287001433270940578127570053434695653912547579202493901871464470\
764410974150819235945558607363556467922279391141836922743541233760821\
871365085257243448039467666652724537269128483841450819722610910449669\
667767218934240840176992473839442233550613463850853485560398554469357\
611537501209062167152021399590356213475289096528949653047033419077429\
299028564676171187099901772701404490557158793484243417114875079809449\
627812693323373523747772814832161656651663374496734064538978915783865\
719064174822864696226655708942852177545994665514849721267519945794503\
019098476165255673401283399976325921222555982464845071082560857852257\
814145566478967899059920204170727945409097495154492869672134459786294\
588119159908191561922643839816897827104225462279731830827250471252105\
289727169676968836275320004859466161656331082081074381176816967370602\
357442772694553727911750007625712888624429904817070697897964886832493\
345528030314492840383966128843659735252693645758721744757267671336699\
697149807573431757428427239466529780150411631632066022100098544057533\
387989402082327607683479023952713064900546185006785478600337149539296\
663361814407606344949829339294479920368454331595871231338484805775737\
339268396124977267103545541977669058589689418363091692719790628856414\
752071432633359549389607990792617571221154257026279901828952182073374\
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Larger values work fine, but exceed the maximum answer size allowed here.

F# (.NET Core), 46 bytes

(Only works up to n = 5, however.)

let rec h=function 0->1|n->(pown n n)*(h(n-1))

Try it online!

TI-Basic, ^{19 17} 14 bytes

prod(seq(I^I,I,1,Ans+not(Ans

sadly 0^0 or an empty list errors, so 0 has to be turned into 1

Raku, 26 bytes

{[*] (0..*Z**0..*)[1..$_]}

      0..*                  # range from 0 to infinity
      0..*Z**0..*           # zip with exponent
     (0..*Z**0..*)[1..$_]   # slice from range 1 to default variable (var passed to function) inclusive
 [*] (0..*Z**0..*)[1..$_]   # reduce on multiplication
{[*] (0..*Z**0..*)[1..$_]}  # block

Try it online!

Perl 5 -p, 27 bytes

26 bytes for code; +1 byte for -p switch

$.*=$_**$_ for 1..$_;$_=$.

Try it online!

Ungolfed (but somehow not any more readable):

for (1..$_) {
    # The built-in variable $. ($INPUT_LINE_NUMBER) defaults to 1
    $. *= $_ ** $_
}
$_ = $.

O, 14 bytes

[[j,;]{nn^}d]*

Try it online!

[                 # Open the array
 [j,;]            # Input range 1..n
      {nn^}       # Push a block to power by self
           d]     # Apply n close
             *    # Product

Pari/GP, 18 bytes

n->prod(i=1,n,i^i)

Try it online!

Clojure, 104 86 bytes

(defn h[n](->>(rest(range(inc n)))(map #(repeat %1 %1))(map #(apply *' %))(apply *')))

Try this version online!

Cut it down to 86 bytes by using the thread-after macro.

Un-golfed:

(defn hyperfactorial [n]
  (->> (rest (range (inc n)))
         (map #(repeat %1 %1))
           (map #(apply *' %))
             (apply *')))

Previous 104 byte answer:

(defn h[n](let[p(fn[c](apply *' c))m(rest(range(inc n)))f(map #(repeat %1 %1)m)q(map p f)](apply *' q)))

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Ungolfed:

(defn hyperfactorial [n]
  (let [ pow      (fn [c] (apply *' c))
         nums     (rest (range (inc n)))  ; '(1 2 3 ... n)
         factors  (map #(repeat %1 %1) nums)
         powers   (map pow factors) ]
     (apply *' powers)))

TI-BASIC, 13 bytes

(TI-84+ only, for randIntNoRep()

max(1,randIntNoRep(1,Ans
prod(Ans^Ans

Takes input as the last expression entered, like so:

0:prgmH
               1
2:prgmH
               4
4:prgmH
           27648
5:prgmH
        86400000

NB: TI-BASIC is a tokenized language; that means the textual representation of bytes can be multiple characters. In fact, e.g., randIntNoRep( is a single character that cannot be edited as one might edit an ASCII string. This program is represented by the following hex bytes (if I translated it right):

19 31 2B EF 35 31 2B 72 3F B7 72 F0 72

Based off of this site. All tokens are one byte, with the exception of randIntNoRep(, which is represented with two bytes, EF 35. 3F represents the newline.

Explanation

randIntNoRep(1,Ans returns a range of numbers from 1 to Ans. This is, as far as I can tell, the shortest way to generate a range. This occupies 5 bytes. For Ans = 0, we get either {0, 1} or {1, 0} depending on the RNG. Since TI-BASIC errors out when trying to compute \$0^0\$, we need to remove the zeroes by taking the max of each element with 1 (max(1,. Even though we compute 1^1 twice for Ans = 0, this does not affect the overall product.

Next, we simply compute the hyperfactorial factors as Ans^Ans, raising the list to the power of itself. After, we take the prod( of this list, returning a single number.

Wolfram Language (Mathematica), 19 bytes

Product[n^n,{n,#}]&

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also as @att mentioned there is a built-in for this...

Wolfram Language (Mathematica), 14 bytes

Hyperfactorial

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Java (JDK), 51 bytes

double a(int i){return i<1?1:Math.pow(i,i)*a(i-1);}

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returning a double saves bytes as you don't have the cast the Math.pow

Same byte-count as the other java answer but using recursion.

Assembly (NASM, 32-bit, Linux), 85 bytes

H:mov ebx,eax
or ax,0
mov ax,1
je z
n:mov ecx,ebx
x:mul ebx
loop x
dec bx
jnz n
z:ret

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The argument to the H function is passed in the eax register. The result is also in the eax register. The input has to be lower than 65,536.

Arn, 5 bytes

Uses the slightly older online version

O«▒¹Ù

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Explained

Unpacked: *{^}\~

         _    Implied variable, initialized to input
        ~    Range 1..
   *…\  fold with multiplication after mapping
  {   block with key of _, initialized to current number
 ^  raise _ to the power of _
} end block

Raku, 17 bytes

{[*] [\R*] 1..$_}

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• 1 .. $_ is the sequence of numbers from 1 to the input argument.
• [\R*] is the "triangular reduction" (scan) of those numbers with the Reversed (and thus right-associative) multiplication operator, producing this list: 𝑛, 𝑛 × (𝑛-1), 𝑛 × (𝑛-1) × (𝑛-2), ..., 𝑛 × (𝑛-1) × (𝑛-2) × ... × 1.
• [*] is the product of those numbers, which contains 𝑛 factors of 𝑛, 𝑛-1 factors of 𝑛-1, etc, as required.

If the input argument is zero, 1 .. $_ is an empty range, [\R*] applied to that range produces an empty list, and [*] applied to that list returns the multiplicative identity element 1.

Attache, 14 bytes

Prod@{1:_^1:_}

Try it online! Composes the Product function with a function which computes k^k for k in 1..n.

Slightly more elegant, 15 bytes: Prod@{_^_}@1&`:

Less characters, more bytes (13ch/15b): Prod«1:_^1:_», or the equivalent 15b program Prod<~1:_^1:_~>.

Gaia, 3 bytes

*†Π

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† vectorizes the operator on its left, integer arguments are implicitly cast to ranges. * is exponentiation and Π the product of a list.

Java (JDK), 51 bytes

n->{var r=1;for(;n>0;)r*=Math.pow(n,n--);return r;}

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Julia 1.0, ¹⁷ 16 bytes

!n=n<0||!~-n*n^n

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-1 byte thanks to dingledooper - I always forget about ~-n

Scala, 43 bytes

1.to(_)./:(1)((a,n)=>a*math.pow(n,n).toInt)

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For comparison, the recursive solution is 56 bytes:

def f(n:Int):Int=if(n>0)math.pow(n,n).toInt*f(n-1)else 1

Python 3.8 (pre-release), 59 bytes

from math import*
lambda n:prod(i**i for i in range(1,n+1))

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Zsh, 33 bytes

for ((a=1;n++<$1;a*=n**n)):
<<<$a

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PowerShell, 51 bytes

param($n)iex((1..$n|%{[math]::pow($_,$_)})-join'*')

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It is rare to see Powershell answers without spaces

MATL, 4 bytes

:t^p

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First MATL answer!

Uses the same range-dup-power-product

Vyxal, 4 bytes

ɾ:eΠ

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Obligatory answer

ɾ              # Range
:              # Dup
e             # Power
Π            # Product

jq, 42 bytes

[range(.+1)]|reduce.[]as$i(1;.*pow($i;$i))

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Ruby, 24 bytes

f=->n{n<1?1:n**n*f[n-1]}

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Wren, 43 bytes

var F=Fn.new{|n|n<1?1:n.pow(n)*F.call(n-1)}

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Alternate:

Wren, 44 bytes

Fn.new{|n|(1..n).reduce(1){|a,b|a*b.pow(b)}}

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CJam, 11 bytes

1ri,{)_#*}%

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How it works

1             e# Push 1
 ri           e# Read input and intepret as an integer, n
   ,          e# Range. Gives [0 1 2 -... n-1] 
    {    }%   e# For each k in the range
     )        e# Add 1
      _       e# Duplicate
       #      e# Power
        *     e# Product
              e# Implicit display

C (gcc), 28 bytes

H(n){n=n?pow(n,n)*H(--n):1;}

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MathGolf, 8 bytes

╒_m#ε*1╙

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Explanation:

╒         # Push a list in the range [1, (implicit) input-integer]
 _        # Duplicate it
  m#      # Take the exponent of the two lists at the same positions
    ε*    # Reduce this list by multiplication
      1╙  # Leave the max of this value and 1 (for edge case input=0)
          # (after which the entire stack is output implicitly as result)

Haskell, 20 bytes

h 0=1
h n=n^n*h(n-1)

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APL (Dyalog Classic), 5 bytes

⍳×.*⍳

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Monadic fork train

First it evaluates range of the input, and dots the power and takes the product.

R, 20 bytes

prod((x=1:scan())^x)

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Jelly, 3 bytes

*)P

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How it works

*)P - Main link. Takes n on the left
 )  - Over each integer 1 ≤ i ≤ n:
*   -   Raise i to the power i
  P - Product

Python 3, 28 bytes

f=lambda n:n<1or n**n*f(n-1)

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JavaScript (ES7), 20 bytes

f=n=>n?n**n*f(n-1):1

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05AB1E, 4 bytes

LDmP

Try it online or verify all test cases.

Explanation:

L     # Push a list in the range [1, (implicit) input-integer]
      # (note: input 0 will create the list [1,0])
 D    # Duplicate this list
  m   # Take the exponent of the two lists at the same positions
   P  # Take the product of this list
      # (after which the result is output implicitly)

Husk, 5 bytes

Πm´^ḣ

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Π       # product of
 m      # mapping across all values of
     ḣ  # 1..input
  ´^    # x to the power of x

Charcoal, 8 bytes

ＩΠＥ⊕ＮＸιι

Try it online! Link is to verbose version of code. Explanation:

    Ｎ       Input `n` as a number
   ⊕        Incremented
  Ｅ         Map over implicit range
      ι     Current value
     Ｘ      Raised to power
       ι    Current value
 Π          Product
Ｉ           Cast to string
            Implicitly print