What's the highest possible amount of free Carry Capacity?

Solution 1:

Below you'll find the best I've come up with. All the armor is from the basic tier. Heavier armors weigh more.

Base Capacity
150 lbs

11 Strength
+110 lbs

Strong Back Perk Rank 2
+50 lbs

Lone Wanderer Perk Rank 2
+100 lbs

Army Fatigues
Weighs 3 lbs, +1 to STR, for +7 to capacity

Baseball Uniform or Dirty Army Fatigues would work just as well.

Fortifying Synth Helmet
Weighs 3 lbs, +1 to STR, for +7 to capacity

Fortifying is a legendary effect that gives +1 to STR and END. SE user MeesterTeem reports having found exactly this helmet.

Fortifying Deep-Pocketed Leather Chest
Weighs 5.5 lbs, +1 to STR, +20 Capacity, for +24.5 to capacity

Fortifying Deep-Pocketed Leather Left Arm
Weighs 2.2 lbs, +1 to STR, +10 Capacity, for +17.8 to capacity

Deep-Pocketed Combat Armor limbs also weigh only 2.2 lbs, and would work just as well.

Fortifying Deep-Pocketed Leather Right Arm
Weighs 2.2 lbs, +1 to STR, +10 Capacity, for +17.8 to capacity

Fortifying Deep-Pocketed Leather Left Leg
Weighs 4.2 lbs, +1 to STR, +10 Capacity, for +15.8 to capacity

Fortifying Deep-Pocketed Leather Right Leg
Weighs 2.2 lbs, +1 to STR, +10 Capacity, for +17.8 to capacity

150 + 110 + 50 + 100 + 7 + 7 + 24.5 + 3(17.5) + 15.8 = 516.8 lbs

Notes

I got all my information for this answer from Nukapedia. The information there isn't complete, but I'm assuming patterns hold. I updated the answer with new information at a later date, and the capacity changed by 0.2 lbs.

I'm not considering any kind of cheating or exploits, like using the You're Special book more than once. If you want to cheat, just use the console. According to the Carry Weight page on Nukapedia:

Using the console command "player.modav" carry weight can be increased to a max of 20,106 lbs.

Nukapedia lists legs as weighing the same, but I've found that one of the leather legs weighs 4 lbs, and the other 2 lbs. I think it was the left one that weighed more? Gotta check that.