In the Iliad, Agamemnon offers Achilles one of his daughter (whichever Achilles wants) with “no bride price, and full dowry.”
Agamemnon is the leader of the Greeks, but he is of slightly lower social standing than Achilles.
He is the leader because he agreed to sacrifice his daughter to the gods to guarantee success victory in war, but this victory will only happen if Achilles joins him.
Agamemnon has tons of daughters and no sons. He is extremely willing to expend his daughters. This may or may not be because he doesn’t want to pay their dowries.
In return, Achilles must recognize Agamemnons kingship, and agree to follow orders.
Achilles knows from divine prophecy if he accepts Agamemnons offer he will die very soon, so he doesn’t really care. The marriage isn’t happening regardless.
But Achilles wants to join the battle, and die, for personal reasons (vengeance) unrelated to Agamemnon.
So the offer of “marriage with no bride price and full dowry” is just a sad pathetic joke.
It’s a way for Agamemnon to seem like he’s in control, and to remind the Greeks why he’s king - because he’s a narcissistic psychopath who will do necessary, but horrible things nobody else will do.
The distinction between bride price and dowry only serves to show Agamemnon is making up the rules and the whole thing is a farce.