Infinitesimals [1][2][3] are probably the best example. Though there's a whole range of theories, the basic version is just a magic number, d, such that d^2 == 0. So (x+d)^3 = x^3 + 3dx^2 + 0, giving us a more "natural" path to get to the notion of derivatives without resorting to limits. They get hard to manage, though, so they haven't really made it into the mainstream.

[1] https://en.wikipedia.org/wiki/Non-standard_analysis

[2] https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis

[3] https://en.wikipedia.org/wiki/Dual_number