As far as I know you can actually cast 5 (or 4 depending on interpretation of bonus action spell rules) in a round if you have taken a level as a wild magic sorcerer and 2 as a fighter. The relevant oddities are

• Fighter's action surge

Starting at 2nd level, you can push yourself beyond your normal limits for a moment. On Your Turn, you can take one additional action on top of your regular action and a possible Bonus Action.

• Roll on the Wild Magic Surge (WMS) table and hit 81-82 (on a d100 roll). There are a couple of conditions that trigger the following:

the DM can have you roll on the Wild Magic Surge table immediately after you cast a sorcerer spell of 1st level or higher.

• Entry 81-82 reads:

You can take one additional action immediately.

So your five spells could look like this:

• Cast a cantrip as an action
• Cast a level 1+ bonus action sorcerer spell spell (can use sorcerer's quicken spell to make an action spell a bonus action spell). Roll on the WMS table and hit 81 or 82.
• Cast a cantrip using this extra action from Wild Magic.
• Cast a cantrip using the action from Action Surge
• Cast a reaction spell (e.g. shield)

Note: If you interpret the bonus action spell rule

You can't cast another spell during the same turn, except for a cantrip with a casting time of 1 action.

to mean you can only cast one other spell during the entire turn, then you can only cast 4 (strike the bonus actions spell from the above list and instead hit the WMS on one of the other two action spells).

As far as I know you can actually cast 5 (or 4 depending on interpretation of bonus action spell rules) in a round if you have taken 1 level as a wild magic sorcerer and 2 levels as a fighter. The relevant oddities are

• Fighter's action surge

Starting at 2nd level, you can push yourself beyond your normal limits for a moment. On Your Turn, you can take one additional action on top of your regular action and a possible Bonus Action.

• Roll on the Wild Magic Surge (WMS) table and hit 81-82 (on a d100 roll). There are a couple of conditions that trigger the following:

the DM can have you roll on the Wild Magic Surge table immediately after you cast a sorcerer spell of 1st level or higher.

• Entry 81-82 on the WMS table reads:

You can take one additional action immediately.

So your five spells could look like this:

• Cast a cantrip as an action.
• Cast a level 1+ bonus action sorcerer spell spell (can use sorcerer's quicken spell to make an action spell a bonus action spell). Roll on the WMS table and hit 81 or 82.
• Cast a cantrip using this extra action from Wild Magic.
• Cast a cantrip using the action from Action Surge.
• Cast a reaction spell (e.g. shield).

Note: If you interpret the bonus action spell rule

You can't cast another spell during the same turn, except for a cantrip with a casting time of 1 action.

to mean you can only cast one other spell during the entire turn, then you can only cast 4 (strike the bonus actions spell from the above list and instead hit the WMS on one of the other two action spells).