How can we compute the mean and variance of $e^{W\_tW\_s} $ where $(W\_t)\_{t \geq 0} $ is a Brownian motion?

If we want to compute $ \mathbb{E}(W\_tW\_s) $, the usual thing to do is to assume that $ s \leq t$ and write $W\_tW\_s = (W\_t - W\_s)W\_s + W\_s^2$, then we can use independence of increments to finish that computation, but due to the exponential function, that trick does not work here unless we can write $W\_tW\_s$ as a sum of independent increments, so that $e^{W\_tW\_s}$ can be written as a product of functions of the independent increments.

Does anyone have any suggestions?