Let me put it this way: there's a reason why nobody posted "there exists a sporadic simple group of order greater than n" as an answer to the "eventual counterexamples" question.]]>

I closed the question as a duplicate, because I didn't think a logically equivalent rephrasing merited a separate question.

]]>115132219018763992565095597973971522401 is the 88th and last n-digit number equal to the sum of the n-th powers of its digits.

73939133 is the 83rd and last right-truncatable prime (every prefix is a prime).

1111111110 is the 84th and last number n equal to the number of ones in the decimal representation of all the numbers up to and including n.

357686312646216567629137 is the last left-truncatable prime (no zeros, and every suffix is prime); there are 4260 such primes.

I'm sure everyone will recognize 808017424794512875886459904961710757005754368000000000 as the 26th and biggest order of a sporadic simple group.

1598455815964665104598224777343146075218771968 is the 36th and last 4-perfect number (the sum of its divisors is 4 times the number).

I don't see any natural way of fitting any of these into the "eventual counterexample" question.]]>