Within the limits of disk storage you could have any number of different apps. Shiny Server runs fine on an Amazon Cloud instance with 1 core and 1 Gigabyte of memory.
The limit is on the number of sessions (users, basically) you can have hitting the server at one time before performance deteriorates. Scaling Shiny is a video that demonstrates Shiny supporting 10,000 sessions simultaneously, although that takes a lot more and bigger Amazon Cloud instances..