Sit down, son.
Machinist: 7306.53 potency done during the duration of Wildfire (including the WF pop).
Dragoon: 6446.59 potency done while Blood for Blood is active.
Neither of these include the 3 (MCH) or 2 (DRG) lead-up GCDs before the period I measured. Both of these end less than 25s into the fight.
BLM is likely higher than either, considering the fact that their Fire IV (of which there are 5-6 in the opener) is 500+ potency without Raging Strikes included.
4800 is garbage.
Calculations:
In the 15s during Wildfire, a Machinist will do:
Slug Shot [HE+B4B+RS+HC+GB+A] 200x1.15x1.1x1.2x1.1x1.3x1.05 = 455.85
Clean Shot [HE+B4B+RS+HC+GB+A+Auto-Crit] 220x1.15x1.1x1.2x1.1x1.3x1.5x1.05 = 752.16
Split Shot [HE+B4B+RS+HC+GB+A] 160x1.15x1.1x1.2x1.1x1.3x1.05 = 364.68
Slug Shot [HE+B4B+RS+HC+GB+A] 200x1.15x1.1x1.2x1.1x1.3x1.05 = 455.85
Split Shot [HE+B4B+RS+HC+GB+A] 160x1.15x1.1x1.2x1.1x1.3x1.05 = 364.68
Slug Shot [HE+B4B+RS+HC+GB+A] 200x1.15x1.1x1.2x1.1x1.3x1.05 = 455.85
Clean Shot [HE+B4B+RS+HC+GB] 200x1.15x1.1x1.2x1.1x1.3x1.05 = 455.85
Slug Shot [HC+GB] 180x1.1x1.3x1.05 = 270.27
Split Shot [HC] 140x1.1x1.05 = 161.7
Total for GCDs: 3736.89
In addition to:
Gauss Round 200x1.15x1.1x1.2x1.1x1.3x1.05 = 455.85
Ricochet 300x1.15x1.1x1.2x1.1x1.3x1.05 = 683.78
Blank 100x1.15x1.1x1.2x1.1x1.3x1.05 = 227.93
Head Graze 100x1.15x1.1x1.2x1.1x1.3x1.05 = 227.93
Total for oGCDs: 1595.49
And 5 ticks of Lead Shot:
45x1.15x1.1x1.2x1.1x1.3x1.05x5 = 512.84
That totals to: 5845.22
Wildfire deals 25% of that: 1461.31
The final total is: 7306.53 potency.
And I didn't include Disembowel or Trick Attack.
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In the 20s of Blood for Blood, a Dragoon will do:
Disembowel 220x1.15x1.3 = 328.9
Chaos Thrust 250x1.15x1.3x1.1 = 411.125
Chaos Thrust dot 35x8.2x1.15x1.3 = 429.065
4th Hit 290x1.15x1.3x1.1 = 476.905
Phlebotomize 170x1.15x1.3x1.1 = 279.565
Phleb DoT 30x8x1.15x1.3 = 358.8
True Thrust 150x1.15x1.3x1.1 = 246.675
Vorpal Thrust = 200x1.15x1.3x1.1 = 328.9
Full Thrust = 360x1.15x1.3x1.1x1.5 = 888.03
4th Hit 290x1.15x1.3x1.1 = 476.905
Heavy Thrust 170x1.15x1.3x1.1 = 279.565
Total for GCDs: 4504.435
While also hitting:
Leg Sweep 120x1.15x1.3x1.1 = 197.34
Jump 200x1.5x1.3x1.15x1.3x1.1 = 641.355
Spineshatter 170x1.3x1.15x1.3x1.1 = 363.435
Dragonfire 250x1.15x1.3x1.1 = 411.125
Geirskogul 200x1.15x1.3x1.1 = 328.9
Total for oGCDs: 1942.155
The final total is: 6446.59
Again, not including Hypercharge or Trick Attack.
I also didn't adjust for the +25% crit chance that these skills all have.
(15% for Dis -> second 4th, 10% for Dis -> VT, 25% on every ogcd)
//EDIT:
WAR is not ~4800, but actually ~6178.41 potency during a properly executed triple-fell-cleave Berserk. Much better than 4800, but still below the two shown above and likely most - if not all - other DPS Jobs. They /might/ be able to beat MNK and NIN burst, but I'd be shocked if they beat a proper BRD, BLM, or SMN opener. I just can't be arsed to calculate all of these.
The triple Fell Cleave Berserk looks like this:
Fell Cleave 500x1.05x1.5x1.2x1.1 = 1039.5
Maim 190x1.05x1.5x1.2x1.1 = 395.01
Storm's Eye 270x1.05x1.5x1.2x1.1 = 561.33
Heavy Swing 150x1.05x1.5x1.2x1.1 = 311.85
Skull Sunder 200x1.05x1.5x1.2x1.1 = 415.8
Butcher's Block 280x1.05x1.5x1.2x1.1 = 582.12
Fracture 100x1.05x1.5x1.2x1.1 = 207.9
Fracture DoT 20x10x1.05x1.5x1.2 = 378
Fell Cleave 500x1.05x1.5x1.2x1.1 = 1039.5
Fell Cleave 500x1.05x1.5x1.2x1.1 = 1039.5
Brutal Swing 100x1.05x1.5x1.2x1.1 = 207.9
Total: 6178.41
NONE of the above takes Auto Attacks into account. MCH has no autos, but they DO have a 160 damage auto from the Turret every 3s for that duration. It doesn't get any buffs besides its own Hypercharge, but all 5 of those hits add on to the Wildfire pop.
DRG AA's are stronger than WAR AA's, by which I mean a delay of 3.44 = 5-6 AA's in Berserk, while a delay of 2.88 = 6-7 AA's in B4B.