Short answer: infinitely many, and the phrase “distinct kinds” is not well defined. It’s like asking “how many kinds of graphs are there?” Answer: all of them.
But there is a useful taxonomy of the main species people actually use. I’ll give you a mental map so your brain doesn’t melt trying to enumerate the category of all chain complexes.
I’ll organize by (1) where the complex comes from and (2) what the cochains live on.
A chain complex is:
\cdots \xrightarrow{d_{n+1}} C_n \xrightarrow{d_n} C_{n-1} \xrightarrow{d_{n-1}} \cdots
with d_n \circ d_{n+1} = 0.
A cochain complex is the same thing but degrees reversed:
\cdots \xrightarrow{d^{n-1}} C^n \xrightarrow{d^n} C^{n+1} \xrightarrow{d^{n+1}} \cdots,\quad d^{n+1}\circ d^n = 0.
Given literally any graded module / vector space with such a differential, congratulations, you have “a kind of complex.” So combinatorially there are absurdly many.
What you actually care about is: what geometric / algebraic structure it’s encoding. That’s where the main families come in.
These are the ones closest to your Čech / simplicial / filtration obsessions.