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The following transformations are achieved, seen, and/or referenced Dragon Ball, either in Akira Toriyama's original manga series, Toei Animation's animated adaptation, within video games produced by Bandai Namco, etc.
The "Super Saiyan Pedigree" column from Daizenshuu 10: TV Animation Part 3 (p. 92)
This enormous ape form is triggered by basking in the light of a full moon. By transforming into a Great Ape, one's own battle power will rise sharply by ten-fold. Only when a Saiyan's eyes absorb the sunlight bouncing off of the moon (a special light known as Bruits Waves) will their tail react and cause the transformation. This transformation only occurs when the Bruits Waves being absorbed exceed 17,000,000 Zeno, which is only possible during a full moon.[1]
The ultimate Saiyan warrior who has exceeded one's own limits. Originally, the existence of this legendary being was held in doubt amongst even the Saiyans. In reality, any Saiyan who has a high battle power beyond a certain level can become a Super Saiyan. There are several crucial elements to becoming a Super Saiyan; one must have a gentle heart, be exposed to an extreme sense of danger, be filled with a strong anger, be tormented with sorrow, and feel as if the fate of the Saiyan race itself is at risk. A Super Saiyan will be embued with a golden aura and hair (which will stand on end), the sharp eyes of a hawk, and emerald green pupils. In addition to the outward change in appearance, the 50-times increase in the Saiyan's ordinary battle power is proof of the legendary strongest warrior's existence.[2]
Super Saiyan 5: thus far does not exist in official Dragon Ball merchandise, but was somewhat-joking referred to off-hand and explicitly named as such in Dragon Ball GT episode...