Yes, the whole paradigm is to use 'reactive' functions - i.e. functions whose output is cached.

I guess I'm wondering it cache across requests?

In the Shiny framework we think less about short-lived HTTP requests, and more about long-lived (websocket) sessions. Reactive expressions cache their results within or across sessions, depending on where they are declared (this is just a natural result of R's lexical scoping).

http://rstudio.github.io/shiny/tutorial/#scoping