Maximum average bonus to check if team supports: +61
- Maximized Charisma: +10 with 5 x Tome of Leadership and Influence
- Maximized Wisdom: +10 with 5 x Tome of Understanding, needs Fey Wanderer Ranger 3 for Otherworldly Glamour which allows you to add Wisdom bonus to Charisma ability checks [credit to @nben]
- Jack of All Trades: +3 (half profciency bonus) needs Bard 2 (you need College of Lore Bard 6 or Bard 10 to get access to Counterspell via Magical Secrets)
- Dark Ones Own Luck +5.5 from +d10, needs Fiend warlock 6 (usable only once per rest)
- Fortune of the Many +3 from Hobgolin race (usable 6 times/long rest)
- Luckstone: +1
- Ioun Stone of Mastery: +1
Team support, some costing their reaction, bonus action, or action.
Inspiration and flash of genius are also limited uses per rest.
- Flash of Genius (reaction): +10 from maxed Intelligence, needs 5 Tomes of Clear Thought, Artificer 7
- Bardic Inspiration (bonus action): +3.5 from +d6 or +6.5 from +d12, needs at least Bard 1, Bard 15 for full
- Emboldening Bond (free, 1/turn while active): +2.5 from d4, needs Peace Cleric 1
- Guidance (action, concentration): +2.5 from +d4, needs Cleric/Druid/Artificer 1
- Bend Luck (reaction) +2.5 from +d4, needs Wild Magic sorcerer 6
- Lost Crown of Belsimer (bonus action) +3.5 from d6, 3 charges
Characters to sanity check reaction/bonus action loadout:
Lore bard 6/Fiend warlock 6/Fey Wanderer ranger 3/any 5 can counterspell
Bard 15/Peace Cleric 1/any 4 (action, bonus, free) can counterspell
Wild Magic sorcerer 6/wizard 14 (reaction, bonus with Crown)
Artificer 7/any 13 (reaction)
Total expected bonus: +10 +10 +3 +5.5 +3 +1 +1 +10 +6.5 +2.5 +2.5 + 2.5 +3.5 = +61
This assumes no epic boons, such as Boon of Luck.
The Lucky feat could also be used for advantage on at least 3 of these checks, net effect depends on spell level, I leave it out as it is limited in uses. Likewise, the Help action by a team member can grant Advantage.
An Abjuration Wizard 10 would get +6 via proficiency from Improved Abjuration [credit to @Eric], but uses Intelligence for Spellcasting, and could not profit from the +10 from Fey Wanderer's Wisdom bonus. Proficiency bonuses also do not stack, so you cannot benefit from both that and Jack of all Trades.
That said, it might be more effective to have a lower bonus, and instead have multiple people counterspell, for example if the opponent has legendary actions to cast more spells. To beat DC 19 reliably, all you need is to get to +18 on the roll, and a max Int Abjuration Wizard gets to +18 by themselves with just magic items. Forgoing +2.5 from sorcerer adds a third reaction available for counterpell. In practice you probably do not even need to go to +18, as you can withold Bardic Inspiration and other boosters for the few cases where you fail the check.