Try using a weapon loadout that is "sub-optimal" for your class, with no weapon mods.
As a Soldier, only have a pistol
As a Vanguard, only have a sniper rifle
As an Infiltrator, only have a shotgun
As an Engineer, Adept, or Sentinel, carry the heaviest weapons you can find, like the Black Widow and Claymore
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Only level up your signature power, passive, and Fitness.
Use only that power.
For an extra challenge, only level up your signature power; no Fitness, no passive.
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Never use your signature power.
Try playing each class without their go-to move.
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Never change your weapon loadout in the Normandy or prior to a mission.
Every time you come across a weapon in the field, you MUST change weapons.
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Melee only.
You can carry weapons (since you must have at least one), but never fire it.
Heavy melee, melee combo, grab-n-stab are all viable, but no weapons and no powers.
To make it easier, you can use melee mods like the shotgun blade or pistol stunner.
Obviously, cutscenes/Interrupts don't count; also events like what happens at the end of the Mars mission, where you have to gun down the Cerberus bot, that's acceptable.
Or when you face down the Reaper on Rannoch, Marauder Shields at the end, a Renegade interrupt where you shoot your gun; things like that won't break the spirit of this challenge.
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Bonus powers only.
No class powers, no "regular" powers, just the power you pick as a bonus.
Carry only one weapon per mission.
For extra difficulty, pick one bonus power from the start and never change it.
If it's an ammo power, use only the Predator pistol.
If it's a grenade power, then you may spend one (and ONLY one) point on a single additional power, but it must be the "weakest" in your class; so things like Pull for Adepts/Vanguards, Cryo Blast for Engineers/Sentinels, Cryo Ammo for Soldiers/Infiltrators.
(Again, what qualifies as "weakest" is up for interpretation, but you get the idea)
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Pick a single weapon.
Use that weapon, and ONLY that weapon, for the entire game.
For an extra challenge, never change your squadmate's loadouts when you acquire them.
Considering the way the game plays out in the beginning, you don't have to start this challenge until you get control of the Normandy after leaving the Citadel for the first time, where you can then use the weapon bench. Of course, you could use the one at the Spectre office, or just choose the Predator, since you get that right from the start.
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Use a "sub-optimal" squad composition.
To clarify, this requires some meta-knowledge, i.e. you already know as the player what you're going to go up against.
The idea is that you pick the squadmates you normally don't select.
Example:
Geth Dreadnought mission, (Tali has to come, assuming she survived), EDI is a good choice, because of her tech powers.
So bring Liara instead, where her biotics are less effective.
Basically, you go against your own thinking. If you want THIS and THAT squadmate for a given mission, you DO NOT pick them, and then pick the ones you think "I don't want to bring them on this mission" instead.
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Well, that's all I can think of at any rate.