If, instead of giving the discrete topology to finite sets $S_i$, I give them the indiscrete topology, (i.e. the only opens of $S_i$ are the empty set and $S_i$ itself), is it possible to take the inverse limit of such a (directed) system $\lbrace S_i\rbrace$?
If yes, what do I get?
You get a subspace of the product space $\prod_iS_i$. The product of indiscrete spaces has the indiscrete topology, so every subspace has the indiscrete topology. Which subspace you get depends on your bonding maps.