Let $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ be the Banach algebra of bounded holomorphic functions defined on the disjoint union of countably many copies of the open unit disk ${\mathbb {D}}\subset {{\mathbb C}}$ . We show that the dense stable rank of $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ is $1$ and, using this fact, prove some nonlinear Runge-type approximation theorems for $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ maps. Then we apply these results to obtain a priori uniform estimates of norms of approximating maps in similar approximation problems for the algebra $H^\infty ({\mathbb {D}})$ .