Find the largest value of power.

Golfscript - 26 chars

~:x,{:b;x,{b?x=b*}%+}*$-1>

Rough translation to Python

x=input()
acc = []
for b in range(x):
    for _ in range(x):
        acc.append((_**b==x)*b) # most of these are zeros
print max(acc)

So it loops way more times that necessary, but that often happens with golfed answers

APL (Dyalog Extended), 13 bytes

⌈/⍳×∘(⊢=⌊)⍳√⊢

Try it online!

⌈/⍳×∘(⊢=⌊)⍳√⊢     Monadic train taking an input n:
         ⍳√⊢     Get the 1st, ..., nth roots of n.
    (⊢=⌊)        Find whether each element equals its floor; i.e. is an integer.
                Returns a list of 0s and 1s.
  ⍳              List of 1, ..., n
   ×            Multiply that with the binary list.
                Now we have a list starting with 1 (since the 1st root of n is n, an integer)
                and larger nonzero values corresponding to the other roots that are integers.
⌈/              Find the maximum.

Haskell, 63 56 characters

Handles ℤ_>0 (56 characters)

main=do n<-readLn;print$last[p|p<-[0..n],b<-[0..n],b^p==n]

Handles ℤ_≥0 (58 characters)

main=do n<-readLn;print$last[p|p<-[0..n+1],b<-[0..n],b^p==n]

Handles ℤ (70 characters)

main=do n<-readLn;print$last[p|p<-[0..abs n+1],b<-[n..0]++[0..n],b^p==n]