A good start might be finite sequences in OEIS. The search keyword: fini returns 4660 results including Gerry's narcissistic number and left-truncatable prime. Another example is A080601 Number of positions that the 3 X 3 X 3 Rubik cube puzzle can be in after exactly n moves with largest term $91365146187124313$.
The search can be automated filtering only large numbers.
Probably you are not interested in this, but artificial solutions can be constructed easily:
Solutions $x$ of $x^2-y^2=\text{big}$ or trivially roots of $p=\prod_{i \in S} (x-i)$.