Jelly, 26 bytes

Lœ?“EgÆ/”vð@“N6XU¥⁴"EɠẒḌḂż

A full program that prints the maximal exponent to stdout, i.e. an integer greater than \$1\$ if a perfect power or \$1\$ if not.

Try it online! - way too slow for its own input (which is no longer a requirement).

Or see the test-suite.

The bytecode is:

4C 1E 3F FE 45 67 0D 2F FF 76 18 40 FE 4E 36 58 55 04 84 22 45 9F BD AD BF F9

which, as base 10 is (TIO)

$$122317173525583340303786060012597342485674060763941233767596025$$ $$ = (5 \cdot 19 \cdot 254279 \cdot 457835662419192582175421)^{2}$$

How?

Lœ?“EgÆ/”vð@“... - Main Link: integer, N > 1
            “... - length 13 string -> S
          ð@     - dyadic chain - f(S, N)...
   “EgÆ/”        - "EgÆ/"
 œ?              - permutation of {that} at lexicographic index...
L                - ...length of {S}
                    -> "ÆEg/"
         v       - evaluate as Jelly code with {N} as the left argument
                     i.e:
    ÆE           -     prime factor exponents (e.g. 500 = 2*2*5*5*5 -> [2,0,3])
       /         -     reduce by:
      g          -       greatest common divisor

If the output really needs to be truthy in our language and falsey otherwise, this, at \$28\$ bytes, prints a non-zero number if a perfect power (truthy) or zero if not (falsey):

Lœ?“ÆgCE/”vɓ@“F6Æ_æṃcSḌɠ2ḌƈÑ

The C produces one minus its argument, and the string is now length fourteen.

Pyth, 29 27 bytes (thanks CursorCoercer)

iFhMrPQssm.6"}ÉoÌ ¦¼ø

Hexdump:

00000000: 6946 684d 7250 5173 736d 2e36 227d 1b82  iFhMrPQssm.6"}..
00000010: c96f cc20 a619 8b99 bcf8 10              .o. .......

Outputs a number >1 for true, 1 for false.

Try it online! - The given input is the program value.

Explanation

This answer cheeses the challenge by using the length of an arbitrary string inside a computation. The string's contents can be varied to search for valid program values. I'm guessing the intent was to disallow putting arbitrary characters inside of a comment, but the challenge seems impossible without some cheese.

Start with the following Pyth program: iFhMrPQ8

     PQ  - Prime factors of the input, e.g. 36 => [2, 2, 3, 3]
    r  8 - Run-length encoding, e.g. => [[2, 2], [2, 3]]
  hM     - Take the first element of each element, e.g. => [2, 2]
iF       - Reduce with GCD

Notice that 8 is used as an "opcode" input to the r instruction - passing values other than 8 results in a different operation. This is important for the irreducibility requirement.

Replace 8 with the following snippet:

     "xxxxxxxxxxxxxx - 14 arbitrary characters in a string
  m.6                - Map each element to 0.6
 s                   - Sum elements
s                    - Convert to int

This is to satisfy the irreducibility requirement: if any part of the string is removed, the length no longer evaluates to 8. If any part of the length-computing code is removed, it also no longer evaluates to 8. It is unlikely that some subset of this string is valid Pyth code evaluating to 8 (i.e. if the string delimiter " is removed).

Then I just used a script to exhaustively check for perfect square program values. I also checked if I could reduce the size of the string literal (for example, by replacing .5 with .6), but no valid programs were found. .6 (and thus a string literal of length 14) is possible with non-printable characters in the string.

The program value is

43307635010343855810373044925567686241308005777350905277347330064

which is the square of

208104865417279121841411971317508

As a bonus, the program is fast enough for it to evaluate its program value in a reasonable time.