We study the non-Markovian Langevin dynamics of a massive particle in a one-dimensional double-well potential in the presence of multi-exponential memory by simulations. We consider memory functions as the sum of two or three exponentials with different friction amplitudes $\gamma_i$ and different memory times $\tau_i$ and confirm the validity of a previously suggested heuristic formula for the mean first-passage time $\tau_{\textit{MFP}}$ . Based on the heuristic formula, we derive a general scaling diagram that features a Markovian regime for short memory times, an asymptotic long-memory-time regime where barrier crossing is slowed down and $\tau_{\textit{MFP}}$ grows quadratically with the memory time, and a non-Markovian intermediate regime where barrier crossing is slightly accelerated or slightly slowed down, depending primarily on the particle mass. The relative weight of different exponential memory contributions is described by the scaling variable $\gamma_i/\tau_i^2$ , i.e., memory contributions with long memory times or small amplitudes are negligible compared to other memory contributions.
