The battle with Red is my favorite boss battle of all time. It's not particularly hard (Even if you're not over-leveled,) but the "setup" for the battle makes it a truly epic experience. I can understand how people might just call it nostalgia, but I think that to a certain extent, it's timeless.
The first time I played through Silver, I didn't anticipate there being anything on Mt. Silver. I had just beaten all the gyms, and thought that this was just some area with hidden items and a few Pokemon that were unobtainable anywhere else. I keep pushing forward, deeper into the mountain, and then suddenly, I realize I'm up really high. I'm up higher than I've ever been in a Pokemon game before. I walk forward, realizing this is the edge of the world... And there's just a single person standing there. It didn't occur to me who he was. I walk up to him and he doesn't say a word. there's nothing left to say. He calmly turns to me, and we fight.
That has never happened in any other video game I've ever played.