Q:

Expectation in infinity

Is there any pair of random variables (X,Y) such that Expected value of X goes to infinity, Expected value of Y goes to minus infinity but expected value of X+Y goes again to infinity? 

A:

Let $H$ take positive values with density $\dfrac{2}{\pi (1+x^2)}$, i.e. a half Cauchy distribution with positive infinite expectation. 
Let $X=2H$ and $Y=-H$ to meet your requirements.