In this paper, we characterize the wavelet compressibility of compound Poisson processes. To that end, we expand a given compound Poisson process over the Haar wavelet basis and analyse its asymptotic approximation properties. By considering only the nonzero wavelet coefficients up to a given scale, what we call the sparse approximation, we exploit the extreme sparsity of the wavelet expansion that derives from the piecewiseconstant nature of compound Poisson processes. More precisely, we provide nearly-tight lower and upper bounds for the mean L2-sparse approximation error of compound Poisson processes. Using these bounds, we then prove that the sparse approximation error has a sub-exponential and super-polynomial asymptotic behavior. We illustrate these theoretical results with numerical simulations on compound Poisson processes. In particular, we highlight the remarkable ability of wavelet-based dictionaries in achieving highly compressible approximations of compound Poisson processes.