I'm trying to remember which Devil said it (knee jerk thought was Scott Gomez) that he understandably couldn't fully appreciate winning the Cup as a rookie.
Devils had a couple younger guys like Brian Rolston and Bill Guerin on their 1995 Cup roster. Rolston made the playoffs eleven times after that and only got out of the first round one more time. Guerin got his second Cup in the twilight of his career. I'd imagine we could dig up some Guerin quotes about whether he thought he'd get another opportunity.
Some guys are wired to try to win as many as possible (looking in Mark Recchi's direction) while other guys weren't interested in being deadline rentals. I vaguely recall Teemu Selanne saying something along the lines that he didn't feel the need to chase the Cup after winning 2007, so that's why he didn't pursue a trade out of Anaheim. I think Milan Hejduk had a similar mentality. Mike Modano also didn't want to be a deadline rental in what he thought would be his final season with Dallas; And then in the offseason he pivoted and signed with Detroit.
I also think of cases like Mario Lemieux's comeback in 2000 when part of his motivation was that his son had never seen him play. Might be an interesting question to pose about how a guy celebrated as a bachelor vs celebrating with a family.
TL;DR - Depends on the player, some guys will want to chase the high whereas others were happy enough to have won it early.