Is there a limit on how much testosterone a trans man can take and still compete?
Or is the level which a trans man would reach always lower than a man?
Yes, it's ways lower, in that they won't have the same advantages as the equivalent man.
This is because once a woman has been through female puberty, the effects that testosterone can have will be limited - you're packing muscle on to a female frame, with female-sized heart and lungs, female bone density, structure, etc.
It's why the East German women could beat all other women, but not the men.
The only place transmen could potentially compete with men is in the sports with a very extreme (low) weight class (eg 50kg or 55kg), because then the men don't have the benefits of lots more strength and size.