Maximum damage is 428
Here's how (I added some things to make the damage more consistent; average is 418.791):
Your Character (let's call him Cody from now on) is a level 20 Goblin (ASs: 10/13/12/18/16/13) with these classes...
- Wizard 10 ---------- [for Empowered Evocation and 2 ASI]
- Cleric/Druid 1 ----- [for Cleric/Druid spell list]
- Sorcerer 4 ----------[for 1 ASI, Extended Spell, and Wild Magic Surge]
- Warlock 1 ---------- [for Hexblade's Curse]
- Bard 2 -------------- [for Jack of all Trades]
- Fighter 2 ----------- [for Action Surge]
... who gets expeditious retreat and shield from his font of magic and carries these items:
- Spellbook containing magic missle and bestow curse
- 6 Spell Scrolls containing contagion (may not need all of them)
- Ioun Stone of Insight (attuned)
- Ioun Stone of Intellect (attuned)
- a wealth of Spell Scrolls containing shapechange
Your Target (let's call her Tessa from now on) is a gas spore because of their exceptionally low Constitution and Wisdom. (although any creature that is not immune to disease or resistant/immune to force, cold, or necrotic is valid if they fail the saves)
- Cody prepares the spells magic missle and bestow curse after a long rest.
- Cody casts shapechange from a scroll with Extended Spell (trying again if he fails the check, 50% chance; attuning to a Luck Blade and/or acquiring the Lucky feat from a special trainer increase the save significantly). This will allow him to transform into a one-eyed shiver (PoTA 207) for the Chilling Mist trait.
If the one-eyed shiver deals damage to a creature in this area, the creature also takes 5 (1d10) cold damage.
- Cody casts expeditious retreat and shield for 2 hours until he rolls 33 or 34 on the Wild Surge Table (restarting from step 2 and resting as necessary).
33-34 Maximize the damage of the next damaging spell you cast within the next minute.
- Cody uses a contagion spell scroll (trying again if he fails the check, one from Action Surge; +7 vs 15 yields a probability over rounds 2-7 is 1 - 35%^7 = 99.94%) to cast Contagion (Flesh Rot) on Tessa; she will fail every save.
Flesh Rot: The creature's flesh decays. The creature has disadvantage on Charisma checks and vulnerability to all damage.
- Cody casts bestow curse (5th level) on Tessa selecting the fourth option; she will fail the save.
While the target is cursed, your attacks and spells deal an extra 1d8 necrotic damage to the target.
- Cody places Hexblade's Curse on Tessa.
You gain a bonus to damage rolls against the cursed target. The bonus equals your proficiency bonus.
- Cody waits for Tessa's flesh to rot.
- Cody casts magic missle with a 9th level slot selecting Tessa as the target for every missle. Tessa takes extra damage from Fury of the Small.
When you damage a creature with an attack or a spell and the creature's size is larger than yours, you can cause the attack or spell to deal extra damage to the creature. The extra damage equals your level.
- Tessa explodes and 2d4 baby Tessa float slowly toward Cody making him question his decision.
f is force, n is necrotic, c is cold
- Each missle has base damage of 1d4 + 1 force
- Each missle deals 6 extra force damage from Hexblade's Curse
- Each missle deals 5 extra force damage from Empowered Evocation
- There are 11 missles
- The spell deals 20 extra force damage from Fury of the Small
- The spell deals 1d8 extra necrotic from bestow curse
- [11*(1d4+12) + 20 f + 1d8 n]
- Tessa is vulnerable to all damage 99.94% of the time from contagion
- [1.999*(11*(1d4+12) + 20 f + 1d8 n)]
- The damage is maximized from Wild Magic Surge
- [1.999*(11*(4+12) + 20 f + 8 n)]
- The creature then takes 1d10 cold from Chilling Mist
- [1.999*(11*(4+12) + 20 f + 8 n + 1d10 c)]
This makes 391.804 force, 15.992 necrotic, and 10.995 cold damage for a total of 418.791 damage on average [Max: 428]