1/97, whose decimal repeats every 96 digits — the most of any two-digit denominator. There's no overall longest, though: larger full-reptend primes always run longer.
For a prime p, the fraction 1/p can repeat up to p − 1 digits, and p = 97 hits that ceiling with a 96-digit block. Primes that reach the maximum are called 'full-reptend primes' because their decimals cycle through every possible remainder exactly once before repeating. But the blocks just keep growing with bigger primes — 1/109 repeats every 108 digits, 1/113 every 112 — so no single fraction holds the record.
