What is y=(y+2)+2 what is the y value?

Let's simplify the equation slightly, to make it easier to identify
the solution if one exists. 

First, we may remove the parentheses on the left side without
changing the equation.
                                     Y = y + 2 + 2

Now, we may add up the 2 + 2 on the right side:

                                     Y = y + 4

Now we can subtract 'y' from each side of the equation:

                                     0  =  4 .

The solution to the equation ... the 'y' value ... is the number that makes this statement true.
                                     0  =  4

Is there any value of 'y' that can make this a true statement ?

I don't think so.  That means that the original equation has no solution.