Conjectures that have been disproved with extremely large counterexamples?

Another example: Euler's sum of powers conjecture, a generalization of Fermat's Last Theorem. It states:
If the equation $\sum_{i=1}^kx_i^n=z^n$ has a solution in positive integers, then $n \leq k$ (unless $k=1$). Fermat's Last Theorem is the $k=2$ case of this conjecture.

A counterexample for $n=5$ was found in 1966: it's
$$ 61917364224=27^5+84^5+110^5+133^5=144^5 $$ The smallest counterexample for $n=4$ was found in 1988:
$$ 31858749840007945920321 = 95800^4+217519^4+414560^4=422481^4 $$ This example used to be even more useful in the days before FLT was proved, as an answer to the question "Why do we need to prove FLT if it has been verified for thousands of numbers?" :-)

My favorite example, which I'm surprised hasn't been posted yet, is the conjecture:

$n^{17}+9 \text{ and } (n+1)^{17}+9 \text{ are relatively prime}$

The first counterexample is $n=8424432925592889329288197322308900672459420460792433$

A famous example that is not quite as large as these others is the prime race.

The conjecture states, roughly: Consider the first n primes, not counting 2 or 3. Divide them into two groups: A contains all of those primes congruent to 1 modulo 3 and B contains those primes congruent to 2 modulo 3. A will never contain more numbers than B. The smallest value of n for which this is false is 23338590792.