How about f(x) = asin(1/x) and g(x) = 1/x? Or are this f and g monotonic?
I guess this will work, but since asin(1/x) is undefined from x = (-1, 1) it technically isn't a function on the reals. It's only a function from R - (-1, 1) to R.
If I come up with no other examples this will do, but I feel there should be an example with a true function from R to R