Solution: Use the identity $\gcd(2^a - 1, 2^b - 1) = 2^\gcd(a, b) - 1$. Compute $\gcd(12, 18) = 6$. Thus, $\gcd(2^12 - 1, 2^18 - 1) = 2^6 - 1 = 64 - 1 = \boxed63$. - Crosslake