Keys Of Traken, I believe. Peter Davison's first story.
The regeneration limit is first mentioned in the Tom Baker story The Deadly Assassin, where it's explained how the Master had used up all of his. Peter Davison's first story was called Castrovalva, and there's no such story as 'Keys of Traken,' methinks you're getting The Keys of Marinus (William Hartnell) and The Keeper of Traken (Tom Baker) mixed up. Sorry, being a bit of a DW Nazi there.
I find the idea of the Doctor being immortal like Jack quite unappealing. It's alright for comers-and-goers like Jack who can serve as a plot device, but having an immortal lead kinda rapes any story of its dramatic potential. Plus, I for one would love to see, in ten or so years' time, the Thirteenth Doctor facing the inevitability of his final death, and having to explain to his companion that there's no regeneration this time. He would, of course, regenerate in the end, giving us a nice mystery to pursue in the following series.