It should be noted that the final version of Chase adds that the player must discard and Advanced Room card to activate his Level Up Ability each turn.
Chase's Level Up Ability can be activated as soon as possible, as it can allow to player to boost Rooms in their dungeon. Because of the condition of the Ability, any cards that allows the player to gain access to Room cards via the Room Deck or the discard pile help this card immensely. Chase is also the only Boss that can prolong the 'damage life' of the Trap Room, The Clock Tower, which comes in the same card set as Chase, as The Clock Tower works on counters that must be removed each turn. And since Chase can add a counter to it each turn, The Clock Tower can essentially keep up its damage.