I have 2000 something traverser stones listed as currency, which I guess means stored with Joachim. If I go to Joachim, he gives me 3. I then spend many more than 3 in Abyssea. I haven't counted, but essentially, whenever I have run out of time I just exit, enter again and request visitant status over the course of many days.
But, occasionally, I get the message that I don't have a traverser stone when I enter Abyssea.
Aside from that, at least one main quest required me to have a traverser stone "in my inventory". Getting Joachim to give me traverser stones seemed to allow the quest to be triggered, though there were of course no traverser stones in my item inventory.
So, when do I need to go get some of the traverser stones and when do more appear without me going to get them?
Right now is a good example. I exhausted all of my time and was ejected maybe 20 hours ago. When I entered again, I was told I had no traverser stones. So, Joachim gave me 3. But within Abyssea I requested 120 minute which used 4 traverser stones. Where did the extra stone come from?