Infinite matrix rings are pretty interesting. I know several interesting examples.
The simplest is the ring of column-finite matrices over a field $F$, which is isomorphic to the ring of linear transformations of an infinite dimensional $F$ vector space. here are some of its properties. There is also, of course, the ring of row-finite matrices, and the row-and-column finite matrix rings. You could also consider the matrices with finitely many nonzero entries, and generate the ring containing those and the identity matrix.
There is also a close-knit cluster of three examples due to Bergman that all live inside infinite matrix rings. Recently, K. C. O'Meara described an algebra "containing" these examples which makes explaining them a lot easier. Here is the description of that algebra. You can find links to the Bergman examples at that link as well.
Personally, I'm curious about infinite upper-triangular matrix rings. I asked a question about a ring like that but haven't gotten any feedback on it.