The function was previously posted at How to handle log[exp(exp(x))-1] , and no CVX formulation was offered. However, that was before @Michal_Adamaszek , the wizard of conic formulation, joined the forum; so he hasn’t had a crack at it yet.
If y is not a CvX variable or expression, you can enter exp(exp(x))-1 <= exp(y). This won’t work if y is a CVX variable or expression.