In the old days of Red and Blue I did a water type run (Blastoise, Vaporeon, Lapras, Omastar, Starmie and Gyarados) and a poison type run (Venusaur, Nidoking, Beedrill, Golbat, Gengar and Weezing) and managed to complete the game both times.
I have tried several times to do it again with different teams but never really got anywhere. I think the best I did was on Emerald with a grass team (Sceptile, Shiftry, Ludicolo, Vileplume, Bellossom and Tropius) I got to about the seventh gym but got bored with it.
At the moment I'm trying to do an "earth" type run through on Y version- Earth meaning rock types and ground types (though not necessarily dual rock/ground) - it was going to just be ground, but then I wonder-traded away my starter (froakie) in the hopes of getting something decent to go alongside the Bunnelby I caught and I ended up with an Omanyte and so decided to go with it. I've just defeated the second gym and my team currently consists of Diggersby, Marowak, Donphan, Steelix, Amaura and Omanyte. (and yes, the Donphan I got as a Phanpy when I wonder-traded away my Charmander, although admittedly there were a few other trades in between before I got a type I could use!)