So we all know that if a fighter action surges he can cast an additional leveled spell, but can a fighter multiclassing as a sorcerer in a sense cast 4 spells by quickening spell twice?
When you Cast a Spell that has a Casting Time of 1 action, you can spend 2 sorcery points to change the Casting Time to 1 Bonus Action for this casting.
The Fighter only has one Bonus Action to use, even after Action Surging, which gets used up with the first Quicken. So the maximum number of spells is 2, keeping to the restrictions on general spell casting:
Bonus Action A spell cast with a bonus action is especially swift. You must use a bonus action on your turn to cast the spell, provided that you haven’t already taken a bonus action this turn. You can’t cast another spell during the same turn, except for a cantrip with a casting time of 1 action.
No - the maximum is 2 spells per round
Normally, you have one action and one bonus action. Action surge gives you an additional action. Quickened spell reduces an action to a bonus action, but you still only get one bonus action per round.
Additionally, Player's Handbook p.202, "Bonus Action", says:
A spell cast with a bonus action is especially swift. You must use a bonus action on your turn to cast the spell, provided that you haven't already taken a bonus action this turn. You can't cast another spell during the same turn, except for a cantrip with a casting time of 1 action.
You can thus cast at most two spells, using either Quicken Spell or Action Surge:
- Cast a spell quickened using a bonus action, and a cantrip with your action (as supported by the above quote) or:
- Use your action to cast one spell, then use Action Surge for a second action to cast a second spell (as supported by the Sage Advice Compendium v.2.3, p.13, "Casting Time"):
If you cast a spell, such as healing word, with a bonus action, you can cast another spell with your action, but that other spell must be a cantrip. Keep in mind that this partic-ular limit is specific to spells that use a bonus action. For instance, if you cast a second spell using Action Surge, you aren’t limited to casting a cantrip with it.
D&D 5e designer Jeremy Crawford gave his opinion on this situation. Crawford's Twitter statements are no longer considered official game rules, though they may help to clarify that this is how the rule was intended:
Q: So you can’t do Spell (Bonus Action), Cantrip (Action), Action Surge: Spell (Action)?
Crawford: You can't.
In another of Crawford's opinions, he says that the p.202 rule on bonus actions means that you can't cast more than one spell on any turn you cast a spell as a bonus action, which, if you accept his opinion (though it's not an official game rule), would rule out the edge case of one quickened spell plus two cantrips:
Casting a bonus action spell means no other spells that turn but a cantrip with a casting time of 1 action.