Up to +162 on any check
For this build I steered away from things like using Tomes to boost all the stats, and consumable items, since this has to be repeatable daily. I'm not even sure I've fully optimized it the way it is, and with those it could definitely be optimized further.
Level 19 Dhampir with the Boros Legionnaire background.
Wizard (Order of Scribes) 13
Fighter (Battle Master) 4
Cleric (Grave Domain) 2
Blinding Smite (3rd, Boros Legionnaire spell)
Mordekainen's Faithful Hound (4th)
Hold Monster (5th)
Skill Expert (Investigation)
Insignia of Claws
Bag of Tricks, Rust
+3 Arcane Grimoire (Attuned)
Amulet of Health (Attuned)
Ring of Spell Storing (Attuned by Simulacrum)
CON 8 [19 with Amulet]
This takes a bit of setup, but can get a very high bonus on any check, and can be done multiple times per day.
Prepare for this procedure by casting hold monster into the Ring of Spell Storing worn by your simulacrum.
Use action to pull creature out of the Bag of Tricks onto the ground. As soon as it comes out, Simulacrum casts hold monster on the creature using the spell stored in the ring. Spell save DC is 22, best creature WIS is +1, so there is no chance of saving.
Simulacrum uses Channel Divinity: Path to the Grave to make the creature vulnerable to all damage from the next attack. You use a bonus action to cast blinding smite using a 4th-level slot, and use Awakened Spellbook to replace the spell's damage type with piercing damage from Mordekainen's faithful hound. As an action, attack with the Dhampir's Vampiric Bite, adding any Battle Master maneuver that increases the damage roll (my favorite is Menacing Attack).
With a +11 to hit and advantage, there is a 1/400 chance to miss (the highest possible AC from the Bag is 12, so only a natural 1 will miss). If this happens, you can use your own Channel Divinity and Action Surge to attempt the bite again on the next turn. If you fail to bite three times (incredibly unlucky), the Channel Divinity wears off, so you'd need to take a short rest and try again. When it does hit, though, the hit is critical due to the paralyzed condition, dealing
(2d4(bite) + 2d8(maneuver) + 1d4(piercer) + 6d8(smite) + 5) * 2 [piercing]
an average of 97 piercing damage (min 32, max 162).
When you attack with the Dhampir's Vampiric Bite ability, you can choose the following option:
You gain a bonus to the next ability check or attack roll you make; the bonus equals the piercing damage dealt by the bite.
You can now make any check you want and (since the damage was all piercing damage) add the entire amount of damage from the bite onto the check. For the purposes of this question, I will be making an Intelligence (Investigation) check since we took expertise in that. With a roll of 20, our investigation check comes out to a minimum of
20 + 12(expertise) + 5(INT) + 32(bite) = 69
and a maximum of
20 + 12(expertise) + 5(INT) + 162 = 199
with an average check of 134. This whole process can be done up to three times per day, using Arcane Recovery to get a third 5th-level spell slot. It could also be done using hold person if there are any tasty looking humanoids around.
NOTE: This build relies upon the ruling that a simulacrum regains expended resources, specifically Channel Divinity uses and Maneuver dice in this case. Answers to this question indicate this is the RAW ruling, but Crawford tweets (which aren't official rulings anyway) indicate RAI is different. Some DMs may rule against this.