Started with a Morse anglo, which I still use when I am travelling. It's light and robust, so I can just toss it into my carry-on luggage, and it's fairly quiet, so I can play in hotel rooms without inducing irate guests and hotel personnel to hammer on the door to get me to stop.
After three years on the waiting list, I got my Wakker A-4 anglo, which is my primary instrument now. A delight to play, extremely well-made, excellent sound.
Michael O'Raghallaigh plays some of his tunes at a lower pitch (3½ to 4 semitones below standard pitch, using standard fingering on an Ab/Eb Jeffries), and I really like the sound of it. So a while back I bought a nice Bb/F Jeffries for playing Irish tunes that sound great when they're played at a lower pitch, especially with the nice growl that you get with Jeffries reeds. Although as Jim Lucas pointed out, you don't make friends at sessions when you show up and start playing tunes two semitones below standard pitch.
Jim was evil enough to loan me a Wheatstone English concertina to play around with, and I'm getting hooked. I'm going to have to start saving up money so I can buy an English at some point. Concertinas can be as addictive as nicotine.