Well, generally in most games, I just can't part with the Pokemon that I've started with, even if it wasn't my starter Pokemon, and even if I don't exactly like that Pokemon as one of my favorites.
For example, I hate Pokemon that don't have my own ID number, and most Pokemon from trades with cruddy nicknames. I also don't like using Charizard that much, compared to the other Gen I starters. However, in Pokemon Diamond / Pearl, I was given a Charmander as a gift from an online acquaintance of mine to help me get started. Ever since then, I just can't part with PEPPER in my team...
Of course, most likely when I find my Leaf Green game, and manage to migrate my Pokemon to the Pal Park to get my legendary bird Pokemon, I'll most likely toss him away in a box somewhere like yesterday's moldy bread. But, um, yeah.
Specifically though, once I get Articuno on my team, I can never part with him. Ever.