As much as you can complete the game in any order, the trainer levels very much encourage a specific order of completion. In this category, let's do the opposite:

1. Defeat all 8 gym leaders in order from highest to lowest level
2. Defeat all 5 titans in order from highest to lowest level
3. Defeat all 5 team star bases in order from highest to lowest level

The three storylines can be done in any order relative to each other, so there's some amount of optimization over when you do each of the three stories.

Time would stop upon reaching the screen where you get the badge and see the picture for whichever of the three paths you choose to finish last (you have to fight the elite 4 in a fixed order, so you can't apply this there), and it can be verified by looking at the map, which shows the order you did each of the challenges.