I will always have a Pidgeot by the end of Gold or FireRed. Pidgey was my first catch! I still remember it. I was leaving New Bark Town after the Rival fight, and I come across a birdy. So I use Tackle twice, and it's almost dead. And, coz I'm not mean, I catch it!
It was my favourite Pokemon of the game, until I headbutted my way to my Aipom, who is now level 100 and easily able to defeat the elite 4 by himself.