Creatures in the MM that have some ability to heal (skip to "Summary of Results" for breakdown:
Aboleth (situational, requires charmed target to heal 3d6, deals same amount to target, aberration CR 10)
Deva (3/day, heals 4d8+2, CR 10), Planetar (4/day, heals 6d8+3, CR 16) and Solar (4/day, heals 8d8+4, CR 21) all these celestials can only heal others.
Couatl (3/day, Cure Wounds, CR 4, Celestial)
Demilich (At will, heals 6d6-18d6 , CR 18, Undead) healing dependent on # of targets failing DC 19 CON save.
Drow Priestess (2 Cure Mass wounds, max 13 Cure Wounds, CR
Flameskull (regains all health 1 hour after death unless holy water, remove curse, or dispel magic is applied to remains, CR 4, Undead)
Clay Golem (regains health equal to acid damage delt to it, CR 9, Construct), Flesh Golem (ditto with lightning, CR 5, Construct), Iron Golem (ditto with fire, CR 16, Construct) only relevant if wrong damage type used.
Hydra (regains 10 HP per head regrown, CR 8, Monstrosity) heads only regrow after a head dies (25 damage in a single turn, no fire damage or heads don't regrow)
Kuo-Toa Archpriest (2 mass cure wounds, CR 6, Humanoid)
Oni (regains 10 per turn, CR 7, Giant)
Revenant (regains 10 per turn unless subjected to fire or radiant, CR 5, Undead)
Sahuagin Priestess (3 mass healing words, CR 2, Humanoid)
Shambling Mound (heals from lightning damage, CR 5, Plant)
Shield Guardian (regains 10 per turn, CR 7, Construct)
Slaad (all varieties regain 10 per turn, CR 5-10, aberration)
Troll (regains 10 unless subjected to fire or acid, CR 5, Giant)
Unicorn (3/day healing touch 2d8+2, Heal Self legendary action (costs 3 actions, 2d8+2), CR 5, Celestial) healing touch only works on others
Vampire (heals 20 unless in sunlight, running water, or subjected to holy water or radiant damage, can also bite [subject to restrictions] for 3d6 healing, CR 13, Undead)
Vampire Spawn (heals 10 under same conditions as vampire, bite can heal 2d6, CR 5, Undead)
Will-O-Wisp (heals 3d6 when killing target with Consume Life, CR 2, Undead)
In addition, Acolytes and Priests can cast Cure Wounds.
Summary of Results
All creatures except the Acolyte, Priest, Sahuagin Priestess, and Will-O-Wisp are at least CR 4, with the vast majority being CR 5 or above. The healing in question is also fairly minor about equal to 1 hit from a fighter at 5th level. The more devastating heals come from boss monsters like the demilich or only heal others like the angels.
In terms of creature type:
Humanoids have the most variety due to spellcasting.
Undead and Aberrations are the most common creature types, followed by Constructs and Celestials.
It is difficult to say what type of campaign will encounter these the most, although evil campaigns are likely to encounter celestials (which are almost if not all good aligned). There is sufficient variety that any campaign could conceivably fight these creatures, most of which will probably serve as a "boss" depending on the level of the campaign.
Note: I only have access to the monster manual, I do not know what can be found in the expansion guides.