The only other thing I recall seeing is just a mild generalization: instead of having the sets actually be subsets of one another, you have a chain of maps:
S_0 \rightarrow S_1 \rightarrow S_2 \rightarrow S_3 \rightarrow \cdots
and then you can take the limit of this diagram:
S_0 & \rightarrow & S_1 & \rightarrow & S_2 & \rightarrow & S_3 & \rightarrow & \cdots \\
\downarrow & & \downarrow & & \downarrow & & \downarrow & &\cdots \\
S & = & S & = & S & = & S & = & \cdots
Of course, this limit won't be unique, since I could replace S with anything of the same cardinality, and just compose all of those downward arrows with a corresponding bijection.
(If all of the rightward arrows are identity maps, then you get something isomorphic to the ordinary set limit)