Looking at it in terms of compound interest seems random. That said, I think the expression lim((1+x/n)^n) is better motivated within Lie theory, since every Lie group admits a faithful linear representation in which the expression lim((1+x/n)^n) makes sense (even if infinite-dimensional Hilbert spaces might be needed, as in the metaplectic group). Then the subexpression 1+x/n approximates a tangent vector to 1.

A nice approach (which is also in Po Shen Loh's preprint) is that this limit can be seen as the Euler scheme for solving the differential equation f' = f. This is related to your viewpoint about Lie theory... maybe slightly more digestible for high school students.