There are 5 regional chests on Mercury. They can be picked up once per character. Opening all 5 gets you an emblem, 5 Mercury tokens, and 2-3 legendary items.

Locations for all 5 chests:

Chest 1

In the Lighthouse, there are 5 books with glowing crosses on them around the area. These have to be pressed in the correct order and can be acted upon by pressing your action button (you will be prompted)

1. On the table to the left of Brother Vance
2. Behind the 2 other disciples on the right side of the room, on a shelf
3. All the way to the left side of the left bookshelves
4. Almost at the far end of the 2 disciples' table, standing up between a stack of other books
5. On the upper half of the bookshelves to the right of the weapons forge. There's a ledge you can stand on to get it.

After this, you can head to the trenched area to the right of Brother Vance, where a conflux will appear. Use your action button on it, and it will open up a little doorway in the opposite trenches that contains the first chest.

Chest 2

At the area where you spawn in when you travel to Mercury, turn around, and take the path that leads to the back of the Lighthouse(not inside, but in the actual patrol area). There will be a Colossus guarding the chest. You can kill him and get the chest or get the chest and not kill him. But those stun rockets are too annoying and overpowered to move past.

Chest 3

To the right of the entrance to the Infinite forest, there is a bunch of Vex plating on the wall that is scale able. It's pretty hard, but when you get up far enough you can see a ledge up to the left you can try to jump up and grab it. It's a little tough. Fair warning.

Chest 4 and 5

Chest four and five can be activated in the same way, but require you to make the public event heroic, and unless you do something like stall on one side of the event, you may have to wait for the next event. They are located on both of the islands you can get boosted to.

Whatever island you get boosted to first is the only island you can activate heroic on. Do the normal steps you can do to activate heroic ( First video that came up on google) and when you reach the top, you can find the chest up there.

Keep in mind whoever puts the arc charges in last on their side is which island you get launched to, so choose your strategy carefully!! For those who didn't see the beginning, you can get an wmblem, legendary loot, and Mercury coins for getting the five chests.

(Taken from https://www.reddit.com/r/LowSodiumDestiny/comments/7ijroc/mercury_regional_chest_guide/).

Pre-Beyond Light

Since the last answer was posted, Destiny 2 as received many updates, seasons, and expansions. As a result, this effectively increased the maximum triumph score. While I was unable to locate a break down of the scores for each category as shown in the previous answer, the highest triumph score I was able to locate was 103k (Reddit thread was posted roughly 1 month prior to Beyond Light release).

Beyond Light

With the release of Beyond Light, the maximum triumph score has drastically changed. Since the expansion archived a good chunk of older content, the maximum triumph score dropped as a result.

While viewing the the triumph screen, you will now see two different numbers:

• Legacy Score: This is the set of numbers on the right. Your legacy score shows the highest triumph score you achieved prior to Beyond Light.

• Active Score: This is the set numbers on the left, formatted as current / maximum¹. The active score is your score that is based on triumphs that are available in Beyond Light.

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¹ Since Beyond Light has only been recently released, the current known maximum triumph score is 16,770 as shown on Bungie's website. When viewing my triumph score, it only displays 16,610. The maximum score that is shown probably increases has you find and complete hidden triumphs.