Can non-metric TSP be approximated within some non-constant value?
Indeed, it trivially can.
If each edge has positive integer weight $w_e$, then any algorithm that outputs a valid tour has approximation ratio $w_{max} = \max_e w_e$, since the optimal solution costs at least $n \cdot 1 = n$ while the cost of any valid tour $T$ is at most $\sum_{e \in T} w_e \le n \cdot w_{max}$. 

Note: If edges can have weight zero this argument won't work, because in that case even deciding if a tour of cost zero exists is NP-hard; you can reduce Hamiltonian Cycle to it. Hence your algorithm will sometime output a nonzero cost tour, when a zero cost tour exists, which means the approximation ratio cannot be bounded.