Let V(n) denote the number of positive regular integers (mod n) that are ≤ n, and let V_k(n) be a multidimensional generalization of the arithmetic function V(n). We find the Dirichlet series of V_k(n) and give the extremal orders of some totients involving arithmetic functions which generalize the sum-of-divisors function and the Dedekind function. We also give the extremal orders of other totients regarding arithmetic func- tions which generalize the sum of the unitary divisors of n and the unitary function corresponding to φ(n), the Euler function. Finally, we study extremal orders of some compositions, involving the functions mentioned previously.