Throughout the Dragon Ball series, Vegeta has managed to keep up with the escalating Super Saiyan transformations of his longtime frenemy and rival, Goku. Every time Goku unlocks a new form, granting himself a significant power boost, a jealous Vegeta trains furiously to close the gap in strength between them - so it has been since the Saiyan's introduction over 30 years ago. However, there is one Super Saiyan form that, canonically, the Saiyan Prince has never transformed into, bypassing it completely to reach higher levels of power: Super Saiyan 3.
Introduced towards the end of Dragon Ball Z, Goku unveiled his Super Saiyan 3 transformation while fighting Majin Buu. Goku revealed that he had discovered the form while training for seven years in the afterlife, granting him a significant boost over Super Saiyan 2. Since then, the only other character ever seen using the form is the composite warrior Gotenks. Before returning to the afterlife, Goku taught Trunks and Goten to draw on the energy created by their Super Saiyan fusion to transform, making it an exceptional case.
But Vegeta, possessing neither access to training in the afterlife nor the vast energy reserves of a fused fighter, was unable to make the transformation himself. In fact, during their rematch, while Goku was back on Earth, he was visibly upset that Goku had kept the form a secret from him. Vegeta would never be seen attaining Super Saiyan 3 in subsequent Dragon Ball series, instead finding ways to skip Super Saiyan 3 entirely.
Dragon Ball GT saw Bulma develop a device that allowed Vegeta to transform into a Golden Great Ape, his Super Saiyan powers augmented by the primal rage of the Oozaru. After regaining control of the towering beast, Vegeta was able to transform into a Super Saiyan 4 instead, bypassing Super Saiyan 3, implying that any Super Saiyan could potentially transform into a Super Saiyan 4 through Bulma's device...if they managed to control their Golden Great Ape state.
While GT is no longer canon to the main franchise, Vegeta would similarly skip Super Saiyan 3 in the actually canon Dragon Ball Super. Super revealed Saiyans as being able to harness godly energy through a ritual to transform into Super Saiyan Gods. While Goku was able to achieve this in a ritual with Vegeta, Gohan, Trunks, Goten, and Pan, it was revealed that Vegeta had also achieved Super Saiyan God, ostensibly after receiving divine martial arts training from Beerus and Whis, Universe 7's God of Destruction and attending angel. Goku and Vegeta would both progress even further with their Super Saiyan Blue transformation combining godly energy with that of a Super Saiyan transformation.
While it's unclear whether Vegeta's transformations into Super Saiyan 4 or Super Saiyan God allow him to attain the lesser form of Super Saiyan 3, the Saiyan Prince appears largely uninterested. Goku explained that maintaining Super Saiyan 3 in a normal, living body burns through energy dangerously fast and creates a tremendous amount of strain. Meanwhile, Super Saiyan God and Super Saiyan Blue evidently do not. That in mind, Vegeta may see the Super Saiyan 3 form as more of a liability than anything beneficial, preferring the additional power of the Super Saiyan God forms without the risks.
