From Dragon Ball Encyclopedia, the ''Dragon Ball'' wiki
The Kaio-ken Kamehameha (界王拳 かめはめ波, Kaiō-ken Kamehameha; Literally meaning "World King Fist Turtle Destruction Wave") is a combination of the Kaio-ken and the Kamehameha techniques and used by Son Goku. Goku transforms into his Kaio-ken form and uses the Kamehameha, inflicting powerful damage depending on what Kaio-ken level he is using for the attack. The only known Kaio-ken levels he used for the attack are Kaio-ken x3 Kamehameha, Kaio-ken x4 Kamehameha, and Kaio-ken x20 Kamehameha (20倍 界王拳 かめはめ波, 20-Bai Kaiō-ken Kamehameha).
This technique looks very similar to a standard Kamehameha. The only noticeable different is that the energy wave is larger and much more powerful. As the Kaio-ken multiplier increases, the energy wave gets larger.
Dragon Ball Z
Main article: Vegeta Saga
After being battered around by Goku's Kaio-ken x3 attack, Vegeta ended up losing his temper. The Saiyan Prince then flew up in the air and launched his Galick Gun, intent on destroying Goku along with Planet Earth. Out of fear of letting the blast hit Earth, Goku used the Kaio-ken x3 Kamehameha to repel the Galick Gun. Seeing how both attacks were evenly matched, Goku used the Kaio-ken x4 Kamehameha, surging through the Galick Gun and blasting Vegeta into the sky. His body, however, was left exhausted and weakened from having used so much power.
Main article: Frieza Saga
Goku was thrown underwater in his battle against Freeza, who attemptted to drown him. After seeing a vision of his friends and family being murdered, however, Goku regained courage and raised his Kaio-ken to Kaio-ken x20, the highest Kaio-ken level. Goku then beat Freeza around and fired a gigantic Kaio-ken Kamehameha at the tyrant, damaging him slightly. It is named Kaio-ken x20 Kamehameha in Dragon Ball Z: Budokai Tenkaichi 3.
The World's Strongest
Main article: Dragon Ball Z: The World's Strongest
Goku used the Kaio-ken x3 attack against Doctor Uiro (which broke off one of his claws) and then launched a Kaio-ken x3 Kamehameha against the giant robotic scientist. The blast destroyed Doctor Uiro's Fortress and sent him into orbit.
Main article: Dragon Ball Z: Cooler's Revenge
Seeing as he was no match for Cooler in his final form, Goku used the Kaio-ken x20 Kamehameha against the tyrant. Unlike his brother, Freeza, who was damaged by the attack, however, Cooler simply surged through the blast and attacked the surprised Goku.
- The Kaio-ken x20 Kamehameha is incorrectly spelled Kai-ken x20 Kamehameha in Dragon Ball Z: Budokai Tenkaichi 3.