So a sufficiently high level fighter. GWM.
4 attacks a round.
Action surge 8.
Cleave 9
Hew 10
Haste 11
Reaction attack 12
Reaction attack cleave 13.
Asking for a friend.
If one cares for multiclassing:
Warlock 1 (or Pact of the Tome via Eldritch Adept if your DM isn't a certain way about its wording) / Fighter 2 / Valor Bard 14+.
You'll want Eldritch Blast from (Warlock / Pact of the Tome), Nick + Action Surge from Fighter, and Warcaster.
Normal Attack Action: 6 Attacks
- Weapon Attack, Nick Attack, Extra Attack -> Eldritch Blast (4 Attacks)
Action Surge Attack Action: 5 Attacks
- Weapon Attack, Extra Attack -> Eldritch Blast (4 Attacks)
Bonus Action: 1 Attack
- Weapon Attack via Battle Magic feature
Reaction: 4 Attacks
- Eldritch Blast via Warcaster
That gets you to a total of 16 possible attacks.
If you have a way to cast a spell as a bonus action (Cartomancer, Quickened Spell via Sorcerer levels or Metamagic Adept, etc.), you can swap the Bonus Action weapon attack for an Eldritch Blast to up the number of attacks by three (for a total of 19).
If you aren't adverse to using spell slots, and really wanted to get more attacks, you could do something like:
Bonus Action: 9 Attacks
- Cartomancer -> Wish -> Scorching Ray (8th Level)
Normal Magic Action: 10 Attacks
- Scorching Ray (9th Level)
Action Surge Attack Action: 6 Attacks
- Weapon Attack, Nick Attack, Extra Attack -> Eldritch Blast (4 Attacks)
Reaction Warcasting: 9 Attacks
- Scorching Ray (8th Level)
For a total of 34 attacks.
Or, if your DM makes Cartomancer cost a spell slot:
Bonus Action: 10 Attacks
- Cartomancer -> Scorching Ray (9th Level) or Quickened Spell -> Scorching Ray (9th Level)
Normal Magic Action: 6 Attacks
- Weapon Attack, Nick Attack, Extra Attack -> Eldritch Blast (4 Attacks)
Action Surge Attack Action: 5 Attacks
- Weapon Attack, Extra Attack -> Eldritch Blast (4 Attacks)
Reaction Warcasting: 9 Attacks
- Scorching Ray (8th Level)
For a slightly less impressive 30 attacks.
The number of attacks per round could technically go even higher, but (as far as comes to mind immediately) not without violating what I feel is within the spirit of the challenge (single entity without any real prep attacking single target). Otherwise, one can get summons and simulacra and various other items and/or spells involved and really pump the numbers.