> Can anyone please tell me what is the size of session?
As much as the size of stuff you put in it, plus some smallish implementation-dependent overhead.
> Can we add unlimited objects to session?
Depends on the implementation. If you have a distributed web server cluster the session data needs to be distributed among the individual servers somehow, so you don't want to make it too big. Check the documentation of your servlet container if it gives recommendations or even imposes limits.
> what happens if the server memory is full?
OutOfMemoryException probably.
In practice: don't put humongous amounts of data in your session. Make some back of the envelope calculations: 100 simultaneous sessions, 1 kB each = 100 kB.