On average, about 234 damage in one round
Using a plain vanilla fighter from the PHB, you pick Str 14 for one of your stats, pick a race like Mountain Dwarf or Half-Orc that gives you +2 to Strength for a starting Stength of 16. Use Half-Orc for Savage Attack (thanks to Someone_Evil for pointing that out).
Pick the combat style Great Weapon Fighting on level 1 and the Battlemaster subclass on level 3.
You have 6 Stat increases or feat picks up to level 17. You pick:
- Strength Increase x 2 for Strength 20
- Mounted Combatant
- Great Weapon Master
- Magic Initiate (pick Hex)
Pick a greatsword as your weapon, and ride a warhorse.
How many attacks do you have? You attack 6 times, as a full attack action with 1 Attack, 2 Extra attacks, repeated once with your Action Surge. You have two action surges, but only can use one per round.
How much damage does an attack that connects deal? Your attacks on average deal a total of 26.83 damage.
- 2d6, reroll 1s or 2s (8.33, due to Great Weapon Fighting)
- +5 from Strength bonus
- +10 from Great Weapon Master
- +d6 (3.5) from Hex cast as a bonus action before attacks
Critials: You get an extra d6 on these from Savage Attacker. You roll a total of 15 d20 (12 attacks with advantage, 3 Lucky dice), and have 1/20th on each to crit, so you add 15/20 * 4d6 from critical damage, or about 15.3 points (with Great Weapon Fighting style).
How likely are you to hit? You are riding a warhorse, and we opt our opponent to be medium sized, so you attack with advantage. You have 6 superiority dice, one to spend on each attack (pick Precision Attack) to add d10 to the roll. You get +11 (your Strength bonus of +5 added to your proficiency bonus of +6) and -5 (from using Great Weapon Master to add 10 damage) to your attack. According to AnyDice against AC 19 your chance to hit is 87.38%.
set "position order" to "lowest first"
output (2@2d20 +d10 +11 -5) > 18 named "advantage"
I think you can save the superiority dice in cases where you naturally roll high, to instead deal d10 extra damage, but I will ignore this for simplicity's sake.
You know what you have to hit, and you can opt to spend your Lucky dice whenever you miss after you see your rolls. I'm not quite sure how to handle this statistically, but I think it would salvage 67.5% of the failed rolls bringing the probability to 95.9%. You have three Lucky rolls, and on average miss less than one of the six attacks, so it might be higher.
Someone_Evil ran monte-carlo simulations for this build, simulating the use of lucky and precision attack only when relevant and then using a damage maneuver when precision wasn't used. That gave an estimate of 191.6 damage. Picking your battles matters. The observed hit rate was 94%.
Total expected damage: In theory 6 * 26.83 * 95.9% + 15.3 = 170 damage. From practical simulations: 192 damage.
Possible Improvements. It may be possible to eke out a few more points of optimization here and there, maybe by taking Archery for the +2 to hit vs Great Weapon Fighting for the +1.33 damage from die rolls, but then you would need another way go get adavantage on your attacks, as Mounted Combatant only works for melee.
As we only have one round, and there are few spells that can be cast as a bonus action, I do not see Arcane Knight being a better solution. The Champion also seems weaker, it is tripling the crit damage, adding 31 points, but losing the boost to hit from superiority dice, theoretically about 36 points, from simulation up to 58.
Going beyond core rules
The OP allows any published books. Using a Bugbear from Volo's Guide to Monsters for a possible 2d6 points from Surprise Attack would lose the Half-Orcs Savage Attacker and net 1d6 (you would need to surprise them, but we may have that with the first round acting first guaranteed). Another improvement from Shinn Ryusei is to pick Str 15, and trade one of the +2 stat increases for the Orcish Fury feat from XGE, for a one-time extra 1d6.
Goodguy5 reports that in the new Mordenkainen Presents, the once-a-turn limit has been lifted on the Bugbear's Surprise Attack. This makes a large difference: pick Bugbear instead of Half-Orc and add +2d6 to all six attacks for an extra 42 * 95.9% = 40.28. Replace 3.125 points on crits from 1d6 Savage Attacker, with 5.25 points from 2d6 Surpise Attack, for another 2.125 and a total of +42.4 damage over the Half-Orc. This on top of the 192 gets you to 234 expected damage.
Possible Improvements: Maybe there is another fighter subclass or combination in auxiliary books that is even better, and more books are being printed. I'm sure if there is or will be a way to improve on this damage, some clever min-maxer will add a better answer.
The OP stated in the comments that the target is a build that deals more damage to than a Meteor Swarm.
To set a baseline for Meteor Swarm, enemies with Dex Save +5, against a 17th level caster with DC 19 have a 35% chance to make their save for half damage. The baseline damage is 40d6 * (65%*100% + 35%*50%) = 140 * 82.5% = 115 damage per target. Against two targets, the fighter must deal at least 230 damage in one turn to beat the wizard.
Note that the wizard in this case is in no way optimized to maximize his damage. For example, if they were an Evocation wizard, they would add 5 from Intelligence for a total of 120 damage per target.
Our build beats the wizard soundly for a single target, and just barely beats a vanilla wizard when adding a second target. It does not beat an Evoker on two targets, who deals 240.
The challenge is stacked against the fighter by introducing a second opponent so the big spell can deal damage twice, and by limiting it to only one round, which plays to the nova ability of the wizard.
You can easily see this is skewed: if we had to beat the wizards on three targets, we would fail by wide margin. For the maximal number of about 800 opponents that could be hit by Meteor Swarm (assuming each needs at least 5x5 foot of space) we'd be looking at more than 92,000 damage. It would be very likely impossible to do as much damage with the fighter in a single turn without preparation.
The strength of the fighter is that he can deal reliable damage every round. Even without Action Surge this fighter could deal about 60-70 damage each round (I did not do the exact calculation here, he's out of superiority dice after the first round). In spite of the class name, the fighter is not the best damage dealer in the game. Look here to see what multi-classed min-maxing can achieve.