In Fired I caught my first Murkrow, and despite everything, it was on my main team from then on, even through the upgraded E4. Of course, he dies from a lv 1 Tackle, or maybe a slight breeze, but he became my favorite Pokemon, so I didn't care.
"The most important thing is to never stop questioning." -Albert Einstein. Xbox Live: AlteraLateralus
I fricken loved my Mawile in my original Ruby play through. So much that I always get one when it's available in a game.
I always use the obligatory region-specific bird, so I guess those, even though I absolutely love Swellow and Staraptor.
I also tend to become peculiarly attached to the "worthless" normal-types that show up at the beginning of the game. Held on to a Linoone in each of the R/S/E games, trained up a Raticate in both FR/LG, and held onto a Watchog in Gen V. My Watchog in White was pretty weak but the one I have in White 2 can actually hold its own fairly well (so far).
I guess Skitty/Delcatty is another good example.
I never realized that Mightyena was weak O.o Mine tend to be formidable, at least in-game.
I was doing an Ice Type run, and I had Delibird with me. I was telling myself "I should get Articuno", but I was stubborn. But, I'm glad I stuck with him. Not a champ or anything, but he was there when I needed someone to take a hit for the team.
I always use the Farfetch'd you can trade for in Red. It always is surprisingly effective, courtesy to early Swords Dance and Slash/Bodyslam. No brainer whenever I restart Red.
Other than that, I have used Butterfree, Pidgeot and Linoone until E4. Except Butterfree is actually pretty devastating as of FireRed. CompoundEyes powder OP. The other two always held there own pretty well too. Pidgeot rocks in the games he's available, even more with Steel Wing and Linoone is pretty strong in RSE with early Headbutt, speed and Normal/Water coverage.