What kind of operations can we do with bits?
If you think about it, there are $2^4 = 16$ possible ways of combining two bits to give a single-bit ouput (AND, OR, NOR, NAND, XOR, ...).  Can you work out what they all are? This is because there are four possible input combinations ($00$, $01$, $10$, $11$) and any subset of those can be mapped to~$1$.  But we could combine more than two bits to give a single bit output&nbsp;&ndash; can you work out how many operators there are that take $k$&nbsp;bits as input and output one bit?

And we're not even restricted to one bit of output: in general, we can have any number of bits of input and any number of bits of output. You mentioned shifts, which map $k$&nbsp;bits to $k$&nbsp;bits, as an example of this.

Overall, there are infinitely many ways of combining bits.