Boy, this will be a fun question to answer. Simply put, depending on how I look at it, each game could be the worst or the best. I'll only give a description to a few just to save space.
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I like how you broke it down into different aspects. I think I will do the same.
Story:
Story is a culmination of factors, including plot, characters, and atmosphere. You could put 'dialogue' in there as well, but I think that is handled under 'characters'.
Plot:
1. ME1
2. (Tie) ME2 and ME3.
All three games had straightforward plots (Stop Saren, Stop the Collectors, Stop the Reapers) but I thought ME1s plot twists were better and more surprising. ME2 and ME3 were about the same in my opinion.
Characters:
1. ME3
2. ME2
3. ME1
ME3 had the best characters and character development in my opinion. Squadmates appeared to have wandered about the ship between missions and talked with other squadmates or NPCs. And for all the hate against 'auto-dialogue', I think it allowed for greater dialogue and conversations between Shepard and NPCs, and allowed for greater character development and emotional investment. Put it this way: I never liked Garrus in ME1 or ME2 until the dialogue fleshed him out in ME3. (Note that I do not think it was perfect; there were flaws in the system but overall it was a good design. And Ashley got shafted).
ME1 was the first game and so was good at the time with some interesting characters and character development, but ME2 was able to expand on on what ME1 introduced. Characters felt more flushed out, with the loyalty missions providing a greater spotlight on each squadmate.
Atmosphere:
1. ME2
2. ME1
3. ME3
Mass Effect 1 had some great atmosphere as it introduced some alien races and planets that made it feel like a big galaxy. Unfortunately, the planets tended to be human-centric or completely empty. ME2 took the large galaxy and gave us a closer look at the galaxy. We got a closer look at the Krogan on Tuchanka, the Asair on Illium, and the Quarians on the flotilla; we got a closer look at some of the places we only heard about in ME1.
Mass Effect 3 was also diverse, but it never seemed as fleshed out as it could have been.
Overall story:
1. ME1
2. ME2 and ME3 (tie)
I think ME1 barely beats out ME2 and ME3 for the overall story, mostly because of the plot and a galaxy that felt large.
Gameplay:
I wanted to break this down into different categories, but I do feel that it matters since it always ends up:
1. ME2
2. ME3
3. ME1
ME1 felt really fresh to me at the time as it gave me a different shooting experience than what I was used to in other games. But coming back to it after ME2 and ME3 is really really hard. It is clunky, weapons and powers are terrible early, and mods are practically pointless early. And on the highest difficulty, the game switches from 'pretty hard' to 'ridiculously easy' once you level up enough and start getting good gear. It does not hold up well.
Classes in ME1 were diverse, but with bonuses powers obtained from other classes, the hybrids tended to outshine the pure classes; for example, the Vanguard with Singularity was a better Adept, and Infiltrator with AI Hacking was a better Engineer.
Biotic powers were fun (and overpowered), combat powers were good, tech powers seemed weak and very situational.
ME1 equipment (Amps/Tools/armors/weapons) only came in three flavors: crap, soon to be crap, best.
ME1 combat was a cover shooter with space powers, but the cover controls were poor and by mid game you never needed them which made all the combat areas feel rather cluttered. ME2 refined the cover mechanics and made them a core gameplay component this time.
The powers in ME2 were more diverse this time with tech powers gaining more usefulness. Each power felt like it served a useful purpose.
ME2 classes felt unique with the introduction of class-unique powers. The levelling system and new bonus powers allowed for greater variety of builds and playstyles. Insanity was tough but fair. Each weapon type served a purpose and each weapon felt unique.
The new levelling system in ME2 also felt better, with each rank giving the power a significant stat boost (as opposed to ME1 where you had only incremental changes until Advanced and Master). The evolution system was interesting as it theoretically allowed for different builds as well, although most of the time one evolution was clearly better than the other. With shorter cooldowns in ME2 coupled with the new global cooldown made for a faster-paced combat with the requirement that you used the right power at the right time. Combat felt more dynamic this way.
ME3 sought to improve on ME2, but I felt some of the changes were half-baked or unrealized, or just plain broke the balance. The new combat roll was fun and useful. The class-specific heavy melee was a nice touch to make the classes feel more unique but ultimately did not really come into play all that much.
Of the two new class-specific powers, only Nova (Vanguard) and Sentry Turret (Engineer) felt unique; everything else was some type of grenade. And the Turret sucked. And Nova combined with Charge was over-freaking-powered.
The new power combo system felt like it may have had potential, but ultimately I thought it was terrible. Powers no longer held a unique roll (such as shield-stripper or crow-control), but instead turned every power into combo primer or combo detonator. This in turn made each class feel less unique. The Adept in particular is particularly not that different from the Sentinel, and Engineer feels very similar to the Infiltrator.
The weapon weight system in ME3 had potential, but it was tied to the global cooldown, making powers cool down way too fast.
ME3 weapons were very diverse, but most sucked. they also (mostly) got rid of the defense modifiers so weak weapons remained weak.
ME3 actually did have some great level design, but the combat mechanics made the gameplay a joke most of the time.
I actually really enjoyed the level design of ME1's plot missions, but the rest of the maps all felt the same.
Anyway, I am rambling at this point.
Overall:
1. ME2. Good story combined with great gameplay.
2. ME3. Multiplayer was awesome.
3. ME1. The overall story is still good, but the gameplay is a chore.