I always considered Symmetric TSP to be inapproximable in general, and thus by extension Asymmetric TSP as well. Once you add the condition of the triangle inequality however, you obtain Metric TSP (which can be Symmetric or Asymmetric), which is approximable (e.g. Christofides algorithm).
However, I'm having doubts after finding the following paper :
An improved approximation algorithm for ATSP
Vera Traub, Jens Vygen (https://arxiv.org/pdf/1912.00670.pdf)
In their paper, there is no mention of Metric TSP, or the triangle inequality. Does this mean that I'm misunderstanding, i.e. Asymmetric TSP is in fact approximable, even without the triangle inequality?
EDIT : As Discrete Lizard pointed out, Metric TSP seems to not just imply (any) TSP respecting the triangle inequality, but rather Symmetric TSP respecting the triangle inequality. This does not change my last question though, and Yuval Filmus' answer is still correct.