How do I increase my Sync Pair's level cap?

In order to increase level caps, you need to complete Chapter 4 first in the story mode. You can then check Explore -> Training Area -> Reach Past Your Limits!. Depending on your Sync Pair's star rating, you'll have a maximum level cap for your character. Here are the maximum levels based on rarity:

3 Star: 30 - > 45 -> 60 -> 75 -> 90

4 Star: 35 -> 50 -> 65 -> 80 -> 95

5 star: 40 -> 55 -> 70 -> 85 -> 100

In order to know which cap you have, open the Team page while you're at the Pokemon Center and click the "Unlock Level Cap" button.You should see a list of all of your Sync Pairs available. If you see a flashing icon with an arrow, it means that you can raise that specific Sync Pair's level cap since you have the items. You can still however check your other characters as well and see what items you need.

The items needed are dependent on your Sync Pair's role. These resources are limited therefore you need to choose carefully before you increase your Sync Pair's level cap. The items needed are variations of Buff Blends, Aid Ades, Tech Tonics, and Gym Notes. You can gather these in the Training section of the game and complete courses.