output of a cascade system with known impulse responses
exp(-2t)u(t) if I remember right is LTI. Which means you want to take the laplace transform of the inputs and the transfer functions and then use multiplication (the equivalent of convolution when we're in the frequency domain). Once you have all three items transformed and multiplied, you would just inverse transform it to get the answer.

That should be far easier than trying to perform convolution.

Review "Exponential Decay" on this table:
http://en.wikipedia.org/wiki/Laplace_transform#Table_of_selected_Laplace_transforms
You'll end up with (1/(s+alpha))^3 in the frequency domain. Then to convert it back look at "nth power with frequency shift" in that same table.