My question is about Y that is discrete and for some random variable X, but if its having a meaning for Y that is continuous then please give that case your attention.
What can I say about $E[E[X|Y]]$?
I know that E[X|Y] is random variable, so It's not trivial case when we calculate expected value of just a number.
And what about $E[E[X|Y]|Y]$? does something like this have a meaning? if it's, then does for some function $g$ (for simplicity, assuming g with suitable range and continuous) it's true to say that: $$E[g(Y)E[X|Y]|Y]=g(Y)E[E[X|Y]|Y]$$ Because of a theorem that I seen: $$E[g(Y)X|Y]=g(Y)E[X|Y]$$