Given $A,B\in SO(3)$, direct matrix multiplication computes $C=AB$ with 27 multiplies. The group $SO(3)$ is a $3$-dimensional manifold. This suggests that direct matrix multiplication, which thinks of elements of $SO(3)$ as 9-dimensional, is not optimal. What is the minimal number of multiplies necessary to compute $C$?
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Off the top of my head, if you represent elements of $SO(3)$ via lifting them to elements of $SU(2)$, a product can be computed with 8 multiplies. But they're multiplies of complex numbers which are 4 multiplies each for 32 in all.
But, we can use Strassen's algorithm to compute the matrix product in 7 multiplies, and Karatsuba to compute a complex product in 3 multiplies for 21 in all.
Of course, elements of $SU(2)$ aren't general matrices, they have a special form. If you write out the four entries, it's clear you can compute the terms with 4 complex multiplies along with some adds and conjugations, so 12 real multiplies in all if we use Karatsuba or 16 otherwise.
Using Euler-Rodrigues form, we can represent the rotation as 4 real numbers. The stated composition formula requires 16 products. But we can get away with only 10 products. First compute $a_1 a_2$, $b_1 b_2$, $c_1 c_2$, and $d_1 d_2$, then use the idea of Karatsuba to get the rest of the terms in pairs. For example, you can get $a_1b_2 + a_2b_1$ by the single product $(a_1 + b_1)(a_2 + b_2)$ and subtracting off the unwanted terms.
There are more representations listed on wikipedia, but I haven't looked into how much a product would cost in those representations.
Keep in mind that fewer products isn't always better: the adds and other things take time too, and different algorithms behave differently with respect to numerical stability.
I was interested in the same question, and found this thread. Clearly, quaternion or SU(2) multiplication takes no more than 12 real multiplications, since complex multiplication takes no more than 3 real multiplications
The question though is if there are similar tricks that can be done to reduce the number of multiplications for rotation matrices, without converting to another representation. In the 2x2 rotation matrix case, it again takes no more than 3 multiplies, by the same reasoning as in complex multiplication. It's possible, in the 3x3 case, to use the cross product to get the last column of C from the first two. This saves 3 multiplications, and uses the fact that C is orthogonal. It doesn't use the orthogonality of A or B though. Can something better be done?