While there are some daily limits to stat-grinding (hunger/amusement/limits on the number of a specific item that can be used per day such as a Dove feast), there's no actual cap on how many stats a lion can gain over the course of its life. It may get very costly and time-consuming to get a 100-300 stat cub to 1k+ while they're still young, but the only limit is the sales you can find and how much time/funds you can put into it.

The best way to stat-grind before kinging an heir would be to use stat toys (Kudu horns, Dry palm leaves, Flamingo legs, Camel toes, Ostrich fluff, Lovebird plumage) and stat food (Egg yolk, Honeycomb, Crocodile/turtle eggs, Small tortoise, Fluffy bunny carcass, etc.), while completing as many cub training exercises as possible, and finishing off with adol training to 100% Once he's aged up into an adult, keep him on the stat food/toy diet and patrol him as much as you possibly can (some people will keep their heirs as submales to patrol until they reach 14.11 years old as they will be aged down to 8 years upon kinging if they're 10.1 or older).

After kinging, the best things to do are keep feeding stat food, play Baoball every day, PvP for skill, level up as often as possible, and improve impression beyond dreamboat as much as possible.

Overall, the only limit is your capabilities.

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