In addition to all the game features in Miniman's answer, there's also methods of improving your initiative that rely on the fact that initiative rolls count as ability checks:
- Inspiration can be used to gain advantage on an ability check of your choice.
- The Guidance cantrip, available to Druids and Clerics (and other classes through Magic Initiate or multi-classing). The cantrip allows adding 1d4 to an ability check of your choice.
- Bardic Inspiration allows adding 1d6 to an ability check.
- Halfling's Lucky racial trait allows rerolling 1's on ability checks.
- The Lucky feat allows you to roll a second d20 whenever you make an ability check; up to 3 times per long rest.
- Rogue's Reliable Talent turns any d20 roll lower than 10 into a 10 on ability checks that let you add your proficiency bonus. Through multiclassing it's possible to have both Reliable Talent and either Jack of All Trades or Remarkable Athletes, which let you add half your proficiency bonus to dexterity checks that don't already allow you to add your proficiency bonus. Adding half or double your proficiency bonus still counts as adding your proficiency bonus, as confirmed in an unofficial tweet by Jeremy Crawford:
For a rogue/fighter or rogue/bard, Reliable Talent does work with
Remarkable Athlete and Jack of All Trades. #DnD
This is also implied by the Proficiency Bonus rules (PHB, p. 173), which state:
Your proficiency bonus can't be added to a single die roll or other number more than once. (...) Occasionally, your proficiency bonus might be multiplied or divided (doubled or halved, for example) before you apply it. For example, the rogue's Expertise feature doubles the proficiency bonus for certain ability checks. If a circumstance suggests that your proficiency bonus applies more than once to the same roll, you still add it only once and multiply or divide it only once.
The rules still use the term added even though they acknowledge the possibility that your proficiency bonus might be halved or doubled.