Here's another challenge: has anyone ever done a playthrough of the series with melee and powers only (no weapon fire)? Not sure that's even possible. I've played through the entire Bioshock series this way (including 1999 mode BSI!), but the ME series might be harder.
You can do it in ME3, with the exception of certain sequences (Eva on Mars, some latches, and the husks+Marauder at the end where you have to shoot).
Should be possible in ME2, although I have only done it for certain missions. There may be a few instances where you have to shoot to continue. The power core in the Derelict Reaper mission is one instance where you can destroy it with powers, but it takes a veeeeeeery long time to do, so you pretty much need to shoot it to destroy it. But that is the exception instead of the rule.
ME1 would be the hardest, probably impossible. Tech powers are the only real damaging powers, but they do not do that much damage. Biotic and tech powers are mostly about crowd control. You melee with the shoot button but you have to be close to the enemy you want to melee. Melee and power damage becomes increasingly less useful the higher the game difficulty. You would have to do it on Casual or Normal difficulty to do it. Immunity-spamming Krogan would be a pain on higher difficulties.
I just know I would never attempt it in ME1. ME2 is certainly possible and I think I could do it. I pretty much have already done it in ME3.