But OK, I'll do it your way:
An interesting class of algorithms is finite-difference approximations [1]. Even more interesting are algorithms for designing such approximations [2]. A key insight to making these algorithms work is that Taylor series approximations can be used to study the truncation errors of discrete approximations to continuous phenomena. However these techniques, by themselves, mainly interest me in the context of the physical problems they are meant to address. Since those problems come from the physical world, I would argue that the algorithms themselves don't satisfy the formal criteria set forth in the original question, of what algorithms god would write down in a book.
[1] See LeVeque, "Finite Difference Methods for Ordinary and Partial Differential Equations".
[2] See Hamming, "Numerical Methods for Scientists and Engineers".