Well, yes. The 'magic' is the nonlinearity. It's because the composition of linear (actually also affine) functions is still just linear (affine). You don't get any additional power by combining many of them - which is also the reason why linear functions are so well understood and easy to work with.

You give sprinkle in a tiny nonlinearity (e.g. x^2 instead of x) and suddenly you can get infinite complexity by weighted composition - which is also the reason why we're so helpless with nonlinear functions and reach for linear approximations immediately (cf. gradient descent).