The number $91216$ is fun, because $9$, $12$, $16$ is a strictly increasing geometric sequence of length at least three. Does there exist a number that is fun in multiple ways?
The problem can be generalized to have bases other than $10$. Then for any base larger than $32$ we can just use the number $(1,2,4,8,16,32)$ and split it in either $(1)$, $(2)$, $(4)$, $(8)$, $(16)$, $(32)$ or $(1,2)$, $(4,8)$, $(16,32)$. I don't know a solution for any other base.
If leading zeroes are allowed, then $100100010000$ always works giving either $100$, $1000$, $10000$ or $1$, $00100$, $010000$. Even if the sequences are supposed to have different lengths, we can use $100001010000100010000100000$ giving either $1000010$, $100001000$, $10000100000$ or $1$, $000010$, $100$, $001000$, $10000$, $100000$. Therefore, I do not allow the use of leading zeroes.