Perl 5, 32 bytes

s/x|^$/{x^x_x}/g while$b++<5;say

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Japt -h, 21 20 18 16 15 bytes

The 5 at the beginning can be removed to be able to take the level as input.

5Æ="\{^_}"¬qUgx

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Outdated Explanation

5Æ="\{^_}"¬qUª'x
5Æ                   :Map the range [0,5)
  =                  :  Assign to variable U
   "\{^_}"           :    Literal string (the "{" is escaped as it's used for string interpolation in Japt)
          ¬          :    Split
           q         :    Join with
            Uª'x     :      Logical OR of U (initially 0) and "x"
                     :Implicit output of last element

Pyth, 17 16 13 bytes

jF+\x*5]"{^_}

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from functools import*;print(reduce(lambda x,y:x.join(y),["x"]+5*["{^_}"]))

C (gcc), 82 bytes

O(o){o--?_(125,O(o,_(95,O(o,_(94,O(o,_(123))))))):_(120);}main(){O(5);}

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plain TeX, 29 bytes

\def~#1x{{#1x_#1x^#1x}}~~~~~x

That outputs what others have output. But if we need the code to be compilable it would be 6 bytes more

\def~#1x{{#1x_#1x^#1x}}$~~~~~x$\bye

Explanation

~ is an active character in TeX, so we can give it a (new) definition.

\def~#1x{{#1x_#1x^#1x}} defines ~ as a macro, so that when TeX sees ~, it does the following:

• Read everything up to the next x, and call that #1 (pattern-matching).
• Replace the whole thing with {#1x_#1x^#1x}

For example, ~ABCx would get replaced with {ABCx_ABCx^ABCx}.

When ~~~~~x is used, #1 is ~~~~, so the whole thing gets replaced with {~~~~x_~~~~x^~~~~x}. And so on.

Once we have the long string, we can print it out to terminal with \message (and ending with a \bye so TeX stops), so \message{~~~~~x}\bye. Or typeset the resulting expression (as a mathematical formula), by surrounding it in $s : so $~~~~~x$\bye.

PowerShell,  44  35 bytes

"'x'"+"-replace'x','{x^x_x}'"*5|iex

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Uses string multiplication to repeatedly -replace xes with the sub- and super-scripts, then output.

Saved 9 bytes thanks to Joey.

SOGL V0.12, 16 12 bytes

 x5{"{^_}”;∑

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Port of Erik The Outgolfer's Python 2 answer

Jelly, 12 bytes

”x“{^_}”j$5¡

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Port of my Python 2 answer.

05AB1E, 13 bytes

'x5F"{^_}"Ssý

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Port of my Python 2 answer.

Python 2, 32 bytes

exec"print'x'"+".join('{^_}')"*5

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JavaScript (ES6), 45 42 37 bytes

f=n=>n>4?'x':[...'{^_}'].join(f(-~n))

Edit: Saved 3 2 bytes thanks to @Arnauld. Specifying 5 still costs me 2 bytes; this 41 40 35-byte version takes a parameter instead:

f=n=>n?[...'{^_}'].join(f(n-1)):'x'

MATL, 21 20 bytes

'x'XJ5:"J'{x^x_x}'Zt

-1 byte thanks to Giuseppe

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05AB1E, 17 bytes

'x5F'x¡"{x^x_x}"ý

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Explanation

'x                  # push "x"
  5F                # 5 times do
    'x¡             # split on "x"
       "{x^x_x}"ý   # join on "{x^x_x}"

Other programs at the same byte-count include

"{x^x_x}"©4F'x¡®ý
'x5F'x"{x^x_x}".:

Java (OpenJDK 8), 179 167 bytes

@Neil port

interface Y{static void main(String[]a){System.out.println(t.apply(1));}java.util.function.Function<Integer,String>t=N->N>0?Y.t.apply(N-1).replace("x","{x^x_x}"):"x";}

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Wolfram Language (Mathematica) - 40 characters

Summarizing 3 best answers here:

40 bytes:

Nest["{"<>#<>"_"<>#<>"^"<>#<>"}"&,"x",5]

41 bytes:

Nest[StringReplace["x"->"{x^x_x}"],"x",5]

44 bytes:

Last@SubstitutionSystem["x"->"{x^x_x}","x",5]