We consider an additive functional regression model where the responses are $N$ {\it i.i.d.} one-dimensional processes $(Y_i(t), i=1, \ldots, N)$ and the $K$ explanatory random processes $X_{i,j}(t)$ for $j=1, \dots, K$ are observed for $t\in [0,\tau]$, $\tau$ being fixed. The coefficients in the model are $K$ unknown functions $t\mapsto b_j(t)$ for $j=1, \dots, K$ and we build nonparametric least squares estimators under several general settings of explanatory processes, for example, continuous or inhomogeneous counting processes. We bound a mean-square type risk of the estimators from which rates of convergence are deduced. Optimality of the rates is established. An adaptive procedure is then taylored and proved to lead to relevant anisotropic model selection, simultaneously for all functions. Numerical illustrations and a real data example show the practical interest of the theoretical strategy.