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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]
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...