The abilities a leveler grants to itself don’t overwrite any other abilities it may have. In particular, they don’t overwrite the creature’s level up ability; it always has that.

Effects that set a leveler’s power or toughness to a specific value, including the effects from a level symbol’s ability, apply in timestamp order. The timestamp of each level symbol’s ability is the same as the timestamp of the leveler itself, regardless of when the most recent level counter was put on it.

Effects that modify a leveler’s power or toughness, such as the effects of Giant Growth or Glorious Anthem, will apply to it no matter when they started to take effect. The same is true for counters that change the creature’s power or toughness (such as +1/+1 counters) and effects that switch its power and toughness.

If another creature becomes a copy of a leveler, all of the leveler’s printed abilities — including those represented by level symbols — are copied. The current characteristics of the leveler, and the number of level counters on it, are not. The abilities, power, and toughness of the copy will be determined based on how many level counters are on the copy.

A creature’s level is based on how many level counters it has on it, not how many times its level up ability has been activated or has resolved. If a leveler gets level counters due to some other effect (such as Clockspinning) or loses level counters for some reason (such as Vampire Hexmage), its level is changed accordingly.

In a two-player game, if you have a level 7 Lighthouse Chronologist and there are no other extra-turn effects, the game will proceed like this (with extra turns in brackets): your turn, your opponent’s turn, [your turn], your turn, your opponent’s turn, [your turn], your turn, your opponent’s turn, and so on.

In a four-player game, if you have a level 7 Lighthouse Chronologist and there are no other extra-turn effects, the game will proceed like this (with extra turns in brackets): your turn, opponent A’s turn, [your turn], opponent B’s turn, [your turn], opponent C’s turn, [your turn], your turn, opponent A’s turn, and so on. The same principle applies to a multiplayer game with a different number of players.

The effects of multiple level 7 Lighthouse Chronologists are cumulative. If you control two of them in a two-player game, you’ll take three turns for each turn your opponent takes.

In a two-player game, if you and your opponent each control a level 7 Lighthouse Chronologist, they’ll essentially cancel each other out and you’ll alternate turns. Since all the turns you’d be taking are extra turns created by these abilities, make sure you know whose normal turn is supposed to be next in case one or both of the Lighthouse Chronologists leaves the battlefield.

In a multiplayer game, if multiple players each control a level 7 Lighthouse Chronologist, extra turns may sometimes be created faster than they can be taken. Keep track of them carefully. If multiple Chronologist’s abilities trigger during the same turn, the player whose turn would show up sooner in the natural turn order will get the first extra turn. For example, say Players B and C each control a level 7 Lighthouse Chronologist. Player A takes a turn. The game will proceed like this (with extra turns in brackets): Player A’s current turn, [Player B’s turn], [Player C’s turn], [Player B’s turn], [Player C’s turn], and so on. Because Players B and C will each add another extra turn after the other player’s extra turn, Player A will not take another turn as long as both level 7 Lighthouse Chronologists remain on the battlefield.