What is an example of a sequence which "thins out" and is finite?

Solution 1:

An example would be the narcissistic numbers, which are the natural numbers whose decimal expansion can be written with $n$ digits and which are equal to sum of the $n$^th powers of their digits. For instance, $153$ is a narcissistic number, since it has $3$ digits and$$153=1^3+5^3+3^3.$$Of course, any natural number smaller than $10$ is a narcissistic number, but there are only $79$ more of them, the largest of which is$$115\,132\,219\,018\,763\,992\,565\,095\,597\,973\,971\,522\,401.$$

Solution 2:

You ask for a sequence that thins out, seems infinite but is finite. I suggest you follow up instead with an open question.

For people who know (or think they know, or think it's obvious) that the primes go on forever you could talk about twin primes. They begin $$ (3,5), (5,7), (11,13), (17, 19), (29,31), \ldots, (101, 103), \ldots $$ Clearly they thin out faster than the primes, but no one knows whether they stop entirely at some point.

If your audience still has your attention you can say that professionals who know enough to have an opinion on the matter think they go on forever. Then say how famous you would be (in mathematical circles) if you could answer the question one way or the other.

If they are still intrigued tell the story of Yitang Zhang's 2013 breakthrough showing that a prime gap less than $70$ million occurs infinitely often. That was the first proof that any gap had that property. The number has since been reduced to 246. If you could get it down to 2 you'd have the twin prime conjecture.