The "golden section search" to find a the minimum (or maximum) of a unimodal function. An actual real-world use case for the golden ratio.

Exponentially Weighted Moving Average filters. Or how to have a moving average without saving any data points..

Some of my classic favorites:

Skiplists: they are sorted trees, but the algorithms are low complexity which is nice.

Boyer-Moore string search is awesome..

Bit twiddling algorithms from Hackers Delight: for example (x &-x) isolates the least significant set bit: basically use the ALU's carry chain to determine priority.

Another is compress from Hacker's Delight. It very quickly compresses selected bits (from a bitmask), all to the right of the word. It's useful to make certain hash functions.

The humble circular queue. If your circular queue is a power of 2 in size, then the number of bits needed for indexes is at least log2(size) + 1. The + 1 is key- it allows you to distinguish between completely full and completely empty with no effort. So:

    Empty: (write_index == read_index)
    Full: (write_index == read_index + size)
    Count: (write_index - read_index)
    Insert: queue[write_index++ & (size - 1)] = data
    Remove: data = queue[read_index++ & (size - 1)];