I am always the advocate that anything is fine so long as it works.
That said.
If your character is telling a story to someone, then you may want to occasionally interject a: "But what happens next?" asked Lucy. Or a: He paused for a moment and filled his glass full to the brim. Well a little over the brim, which then stopped the story entirely until Benjamin could get him a roll of paper towel. When they had finally cleaned up the mess, Josh continued . . .
Or even you may want to consider: "And then the dragon was upon us!" "What did you do?" Josh leaned over the table and explained in great detail the battle that followed, how he had thought himself a dead man until that strange cloaked figure had entered the scene. "Who was it?" asked Lucy breathless. - see how instead of writing in first person all about the battle ie "I grabbed at his leg, but he was too quick for me, so then I had to dive out of the way as he . . ." which could just go on and on, you can get to the heart of the story much more quickly summing stuff up in one sentence of "and explained in great detail the battle that followed"
You see what I mean? The thing about exposition is that it can get tiring really quickly. However it also depends on the manner of the exposition. If the person telling the story is telling a story and not just a list of facts. If the story itself is entertaining and told well, then you should have no problem.
Good luck!