With Tasha's Cauldron of Everything, 24 tool proficiencies
If you really want to maximize tool proficiencies, it'll involve a lot of multiclass levels. You'll want to make sure you have at least a 13 ability score in each of Charisma, Wisdom, Intelligence, and Dexterity. This should be feasible with the standard array or with point buy, given the new rules in Tasha's Cauldron of Everything: Customizing Your Origin -> Ability Score Increases for applying the Dwarf (Mountain)'s Ability Score Increase to the appropriate ability scores.
(However, a straight-classed Dwarven Artificer can still get access to 13 tool proficiencies by level 4, which is still a lot without any silly ability score spreads or even any need for multiclass rules to be allowed.)
At Level 1: 12 tool proficiencies
Race (7 tool proficiencies)
Choose the Dwarf (Mountain) race. It grants Tool Proficiency (1 tool proficiency), Dwarven Combat Training (4 weapon proficiencies), and Dwarven Armor Training (2 armor proficiencies). Use the rules in Tasha's Cauldron of Everything: Customizing Your Origin -> Proficiencies to swap each weapon and armor proficiency for a different tool proficiency. You can choose any tool, making these true "wildcard" tool proficiencies.
Note: don't choose smith's tools, thieves' tools, tinker's tools, disguise kit, forgery kit, or the last gaming set if you're following this build the whole way!
- Smith’s tools, Brewer’s supplies, or Mason’s tools.
- Wildcard
- Wildcard
- Wildcard
- Wildcard
- Wildcard
- Wildcard
Background (2 tool proficiencies)
Use the rules in Player's Handbook: Backgrounds -> Customizing a Background to customize any background by choosing two tool proficiencies from the sample backgrounds. Note that these aren't full "wildcard" tool proficiencies because not every tool is represented in the sample backgrounds.
Note: don't choose smith's tools, thieves' tools, tinker's tools, disguise kit, forgery kit, or the last gaming set if you're following this build the whole way!
- Disguise kit, Forgery kit, One type of gaming set, Thieves' tools, One type of musical instrument, One type of artisan’s tools, Vehicles (land), Herbalism kit, Navigator’s tools, Vehicles (water).
- Disguise kit, Forgery kit, One type of gaming set, Thieves' tools, One type of musical instrument, One type of artisan’s tools, Vehicles (land), Herbalism kit, Navigator’s tools, Vehicles (water)
Bard (3 tool proficiencies)
Taking your first class level in Bard grants a tool proficiency with three musical instruments of your choice..
- One type of musical instrument
- One type of musical instrument
- One type of musical instrument
At Level 2: 14 tool proficiencies
Artificer (2 tool proficiencies)
Taking your second class level as a multiclass into Artificer grants two tool proficiencies.
- Thieves' tools
- Tinker's tools
At Level 3: 15 tool proficiencies
Cleric (Forge) (1 tool proficiency)
Taking your third class level as a multiclass into Cleric grants access to a divine domain. Choosing the Forge domain grants a specific tool proficiency.
At Level 5: 16 tool proficiencies
Artificer (Any) (1 tool proficiency)
Taking your fourth and fifth class levels in Artificer and choosing to become a specialist in any of the options in Tasha's Cauldron of Everything grants you one additional tool proficiency. It will be a specific set of artisan's tools according to your subclass, except that if you already have that proficiency it instead becomes a proficiency in one type of artisan's tools.
- Alchemist's supplies, Smith's tools, Woodcarver's tools, or One type of artisan's tools
At Level 6: 19 tool proficiencies
Skilled (3 tool proficiencies)
Taking your sixth class level in Artificer grants you an Ability Score Improvement, for which you can instead choose a feat. Choosing the Skilled feat grants three tool proficiencies of your choice.
- Wildcard
- Wildcard
- Wildcard
At Level 9: 22 tool proficiencies
Rogue (Mastermind) (3 tool proficiencies)
Taking your seventh level in Rogue grants proficiency in thieves' tools, but it is wasted since you already knew how to use them from your Artificer levels. Taking your eighth and ninth class levels in Rogue and choosing the archetype Mastermind grants three tool proficiencies.
- Disguise kit
- Forgery kit
- One gaming set of your choice
At Level 12: 23 tool proficiencies
Monk (Drunken Master or Kensei) (1 tool proficiency)
Taking your tenth, eleventh, and twelfth class levels in Monk and choosing the monastic tradition of Way of the Kensei or the Way of the Drunken Master grants you one additional tool proficiency.
- Calligrapher's supplies, Painter's supplies, or Brewer's supplies
At Level 15: 24 tool proficiencies
Fighter (Battlemaster) (1 tool proficiency)
Taking your thirteenth, fourteenth, and fifteenth class levels in Fighter and choosing the archetype Battlemaster grants you one additional tool proficiency.
- One type of artisan's tools
Unfortunately, I am unaware of any new tool proficiencies granted by additional feats you're eligible for, or by any other classes or subclasses. Thus, we have capped out at 24 tool proficiencies)!
Example Proficiency List
- Mason's tools
- Leatherworker's tools
- Weaver's tools
- Carpenter's tools
- Poisoner's kit
- Navigator's tools
- Cartographer's tools
- Vehicles (land)
- Vehicles (water)
- Tinker's tools
- Thieves' tools
- Flute
- Lute
- Lyre
- Smith's tools
- Woodcarver's tools
- Alchemist's supplies
- Cook's utensils
- Jeweler's tools
- Disguise kit
- Forgery kit
- Dragonchess set
- Painter's supplies
- Calligrapher's supplies