Additive rotation matrices

There are no such 3D rotations.

Assume contrariwise that for certain rotations $R_1,R_2,R_3$ the equation $$ R_1\vec{x}+R_2\vec{x}=R_3\vec{x}\qquad(*) $$ holds for all $\vec{x}\in\Bbb{R}^3$. If this works for the triple $(R_1,R_2,R_3)$ then multiplying $(*)$ from the left by $R_3^{-1}$ we see that it also works for the triple $(R_3^{-1}R_1,R_3^{-1}R_2,I_3)$. So without loss of generality we can assume that $R_3$ is the identity mapping.

But $R_1$ has an axis (or $\lambda=1$ is one of its eigenvalues), so there exists a non-zero vector $\vec{u}$ such that $R_1\vec{u}=\vec{u}$. Plugging in $\vec{x}=\vec{u}$ shows that $R_2\vec{u}=\vec{0}$. This is impossible, because as a rotation $R_2$ is non-singular.