Kolmogorov Complexity of the composition of two computable functions
Your bound is correct, for exactly the reasons you give. It is also unimprovable in general. Suppose that each function is multiplication by a large constant, where both constants are subwords of some infinite incomprehensible sequence. If you could compress the composition, you would be able to compress at least one of the constants—a contradiction. 

Edit: You might need to pay additional bits for the composition, as for ordered pairings -- as observed by  Emil Jeřábek in the comments.