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1.
http://mathoverflow.net/questions/101463/properties-of-natural-numbers-such-that-there-is-a-very-large-largest-number-wi was recently closed as a duplicate of http://mathoverflow.net/questions/15444/the-phenomena-of-eventual-counterexamples. I don't see it. The earlier question asked for situations where the first example is large; the newer question asks for situations where the last example is large. It's no more a duplicate than the Ascending Chain Condition is a duplicate of the Descending Chain Condition. Here are a few answers I would contribute, were the question re-opened:

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.
2.

As Joel David Hamkins noted in an answer, you can pass between eventual counterexamples and properties with finitely many examples using a single quantifier and an inequality sign. For example, the property P(n) = "there exists a sporadic simple group of order greater than n" has an eventual counterexample in 808017424794512875886459904961710757005754368000000000.

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

3.
Scott, this logical equivalence misses the point. In the example you give, every number bigger than that one is a counterexample, and that will happen for all "eventual counterexamples" constructed from largest examples. But most eventual counterexamples are not of that kind. E.g., the first counterexample to "cyclotomic polynomials have no coefficients exceeding 1 in absolute value" is 105, but it's not true that every number exceeding 104 is a counterexample; indeed, there are infinitely many examples beyond 104, including every prime and every product of 2 primes.

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.