No FFT? No simplex algorithm? Hashing? Strassen's?
In addition to several algorithms already mentioned, I feel that suffix trees and suffix array algorithms should be there as well. They are making all kinds of approximate searches feasible in bioinformatics.
A cool 19th century algorithm is Radix Sort, though.
All of signal processing went from, "We'd like to do a Fourier Transform but can't afford O(n^2) so here is a faster alternative that kind of does something useful" to, "We start with a FFT."
Neural nets opened a lot of doors.
Bellman equation is in a lot of equipment.
Also I don't really think taking a Taylor series for the inverse square root should count as an "algorithm."
It's an efficient way for performing Lasso (L^1-penalization) to regression models, which has the benefit of (in addition to reducing risk of overfitting) producing sparse models.
SIAM put out a 'ten algorithms of the century' https://www.siam.org/pdf/news/637.pdf a few years ago and it's really tough to argue with six or seven or eight of them.
(MCMC, simplex, Krylov, Householder decomposition, QR, Quicksort, FFT, Ferguson&Forcade's integer relation stuff (that led to stuff like PSLQ), and fast multipole)
And FORTRAN.
