Suppose that user receives reputation only by upvotes and downvotes of her/his question and that for every upvote she/he can have either $0$ or $1$ or $2$ downvotes, with equal probability, equal to $\dfrac {1}{3}$. Also, suppose that downvotes can only be received after an upovote had been received, so that reputation is strictly increasing by $5$, $3$ or $1$ points at every step.
How many prime numbers (as the score of the reputation) are expected (on average) until the reputation is $\geq 100 000$?
As clarified the reputation advances with each upvoted question by $1,3,$ or $5$ with equal chances.
An exact computation is possible using the linearity of expectation.
That is, the expected number of primes below (say) $100,000$ that will be "hit" by a reputation "trajectory" is simply the sum of probabilities of hitting for each such prime.
As Comments by Henning Makholm and lulu explain, the probability of any particular number being hit approaches one-third fairly quickly. So a decent approximation to the expected number of primes occurring in the reputation history is one-third of these primes, e.g. roughly $\pi(10^5)/3 = 9592/3$ before reputation passes $100,000$.