A well-known probabilist once told me that the 3 main classes of stochastic processes are Gaussian, Markov, and martingales.
Martingales are definitely useful in finance and also with respect to other betting games, but it seems to me that they are essential mostly because we have such powerful tools to study them. These tools have often proved useful in much broader contexts, for example the martingale problem/diffusions-- but in my own humble opinion, Gaussian and Markov processes are more natural.
As for Markov processes- the Markov property is clearly very natural. But again IMHO, these processes are most useful in modeling short time behavior, before central limit behavior has kicked in. Although, again stationary distributions and long-time behavior of Markov processes absent Gaussianity are definitely well worth studying (and of course the fruitful study of the mixing of Markov chains connects short term to long term behavior)
Gaussian processes are mostly used to study correlations and dependence, and one might argue that the notions of independence and dependence are what really make probability different from other fields (every remark has its caveat, and the connection between correlations and positive definite functions via Bochner's thm allows analysts to get in this game). Since anything with finite variance converges in a sense to a Gaussian, the idea is that Gaussian processes allow us to study how dependence behaves in the long run (for infinite variance, one simply replaces Gaussian with stable).