What do you get for winning a Legendary Adventure war in CK3?

The attacker gains the contested Title (patch 1.20.0.3).

What does the Legendary Adventure casus belli cost?

piety and prestige; the amount depends on the titles, claims and distance, worked out when you declare.

Casus belli · Conquest

Legendary Adventure

Legendary Adventure is a casus belli in Crusader Kings III (conquest). If the attacker wins: The attacker gains the contested Title. If the defender wins: The defender keeps the contested Title. Declaring it costs piety and prestige. The war goal is worth 80% war score. Patch 1.20.0.3, from game/common/casus_belli_types/06_ce1_wars.txt.

Quick answers (3)

What do you get for winning a Legendary Adventure war in CK3?: The attacker gains the contested Title (patch 1.20.0.3).
What happens on a white peace?: The defender keeps the contested Title.
What does the Legendary Adventure casus belli cost?: piety and prestige; the amount depends on the titles, claims and distance, worked out when you declare.

How the war ends

Attacker wins

The attacker gains the contested Title.

White peace

The defender keeps the contested Title.

Defender wins

The defender keeps the contested Title.

If the war becomes invalid

The … has ended: the defender no longer … the County.

Who can declare it

Unlocked via decision

Every conquest war (the group's allowed_for_character) (7 lines of game script)

herders_and_tributary_constraints = yes
pam_personal_pacifism_offensive_war_ban_trigger = yes
tgp_dynastic_cycle_offensive_wars_ban_trigger = yes
tgp_war_ban_all_ministers_trigger = yes
NOT = {
	government_has_flag = government_is_theocracy
}

As written in game/common/casus_belli_groups/00_casus_belli_groups.txt.

Attacker (allowed_for_character) (9 lines of game script)

custom_tooltip = {
	text = legendary_adventure_cb_tt
	exists = var:legendary_adventure_cb
	var:legendary_adventure_cb >= 1
}
highest_held_title_tier >= tier_duchy
NOT = {
	government_has_flag = government_is_landless_adventurer
}

As written in game/common/casus_belli_types/06_ce1_wars.txt.

Defender (allowed_against_character) (21 lines of game script)

OR = {
	is_ai = yes
	any_sub_realm_kingdom = {
		count >= 2
		save_temporary_scope_as = player_controlled_kingdom
		custom_description = {
			text = SUPPRESS
			scope:defender = {
				completely_controls = scope:player_controlled_kingdom
			}
		}
	}
}
NOT = {
	is_liege_or_above_of = scope:attacker
}
scope:defender = {
	NOT = {
		character_is_land_realm_neighbor = scope:attacker
	}
}

As written in game/common/casus_belli_types/06_ce1_wars.txt.

Cost to declare

Costs piety and prestige. The amount is worked out when you declare (by the titles targeted, claims, distance and more), so the script is shown as written:

Cost (8 lines of game script)

piety = {
	value = fp1_varangian_adventure_base_piety_cost
	multiply = common_cb_piety_cost_multiplier
}
prestige = {
	value = varangian_adventurer_cb_cost
	multiply = common_cb_prestige_cost_multiplier
}

As written in game/common/casus_belli_types/06_ce1_wars.txt.

Source: game/common/casus_belli_types/06_ce1_wars.txt and the English localisation, Crusader Kings III 1.20.0.3. “The attacker”, “the defender” stand where the game names the character; “…” marks other words it fills in during play.