Boss battles: sixty seconds, every four hours
Every four hours a boss appears on the main screen. You have one minute to bring it down, and the only weapon is the same thing you do the rest of the time: tapping. There is no loadout and nothing to buy at the door.
Five bosses exist and you meet them in order — the first fight is the easiest, and from the fifth onward every boss is a fifth-tier boss.

Its health is a share of what you can land
A boss does not have a fixed number of hit points. The game works out what you could land in sixty seconds — your tap power at eight taps a second, which is the rate everything else in the game is priced against — and gives the boss a share of it: three tenths for the first tier, rising to seven tenths for the fifth.
This replaced a health figure that grew with your level. Level comes from experience and damage comes from tap power, and nothing tied the two together, so the boss was growing against the wrong number: a player at level forty with tap power forty could not beat the first tier, while the fifth wanted a tap power of eighty-eight. Tying the health to the damage instead means the fight is always close, and that a point spent on tap power is felt here immediately.
The margin between the share and the whole minute is where the fight actually lives. It is not a clean sixty seconds of hitting: each tier takes five to twenty per cent off every hit, a shield phase takes seventy per cent off while it holds, and a rage phase takes another tenth.
What a defeat pays
A defeated boss pays minutes of your own income: two for the first tier, three, five, eight, and twelve for the fifth. There is a small floor of 200 points so that a beginner’s first fight is worth something rather than a handful of points.
The tiers used to pay flat figures — five thousand up to a hundred thousand — written when a whole day was worth a few thousand points. Left alone they would have become the only thing worth doing. Minutes keep a tier-five boss worth twelve minutes of play for everybody, whatever their minute is worth, which is the same rule behind the daily missions and the events.
Two things multiply it, and both are bought late: Double Boss Rewards and Triple Boss Damage sit in the late shop, behind prestige ten and prestige twenty respectively.

If the connection drops mid-fight
A fight is one minute long and the game holds it open for three minutes after that. Come back inside those three minutes and the fight is still yours, with the damage you had already done intact.
That window exists because a minute is short enough for a dropped connection to cost a whole fight, and a boss that punishes a bad signal is not a boss that tests anything. Outside the window the encounter closes and the next one is four hours away.
One of the three weekly challenges asks for three bosses in the week, and one of the three daily events asks for one — so a boss you were going to fight anyway usually pays twice.