Super Saiyan 3 is the strongest Super Saiyan form in the manga
Super Saiyan 3 is the strongest of all Saiyan forms in the manga. This form is unique compared to the other forms because of the fact that it doesn't require intense emotion to transform, only increased ki capacity. It is technically a forced transformation, like Super Saiyan 2nd and 3rd Stage, but its substantial increase allows it to be considered a true transformation.
Where the 3rd Stage Super Saiyan's purpose was 100% utilization of physical stamina, the purpose of the Super Saiyan 3 transformation is to increase the utilization of ki, and as a result, the transformation rapidly consumes the energy of the user, which leads to extended levels of fatigue, even long after powering down. (If the user has died and isn't encumbered by a living body, this strain is reduced somewhat, and additional energy depletion is minimized.) In Goku's case, however, we can infer that the fatigue may only be due to the fact that he hadn't had much time to practice the transformation, which he stated in his initial demonstration of the form to Majin Buu. The difference between Super Saiyan 3 and Super Saiyan 2, its predecessor, are immediately noticeable as speed, strength and endurance are all pushed past normal limits. The first time that Goku used this form on Earth, it's power caused tidal waves and hurricanes and nearly shook the planet apart. (His subsequent transformations to the form were controlled enough to prevent such dramatic side-effects.) In addition, this power could be sensed even as far as the distance between Earth and Supreme Kai's planet.
However, with the enormous power, there are some serious setbacks. For example, when Goku showed Babidi and Buu the form, he cut his remaining temporal revival energy in half, and his third usage that was present only in the anime (as it was implied that he first achieved Super Saiyan 3 while training on Grand Kai's planet) had him return to Other World completely (in the manga however, Baba only told Goku that he had half an hour left, and he simply spent the remainder of his time before traveling back). Likewise, when Super Saiyan 3 Goku attempted to regain energy while Vegeta and Mr. Satan distracted Kid Buu, he wasn't able to recover the strength needed, thus draining his power completely and leaving him in a weakened, normal state where he couldn't even control the Spirit Bomb until his power was restored by the Namekian Dragonballs. Goku himself had little experience with the transformation, and performing it twice in a short time brought him to his knees and expended all of his remaining energy. (In the manga, he does not revert to his original form until after Vegeta intervenes to buy him the minute that he needed, however.) The time is reduced even further when he is turned back into a child in Dragon Ball GT, where he can only hold the transformation for a couple of seconds. This limit was lifted somewhat after Goku's tail was restored with the help of Elder Kai, allowing him to sustain the form for a longer duration, though he was still no match for Super Baby Vegeta in this state.The video games Vegeta and Broly have attained super sayian 3.
The transformation causes stark changes in physical appearance. The trademark characteristic of the transformation is the hair: the rigid hair of the Super Saiyan 2 state becomes flowing and smooth again, and grows down to or sometime passes the user's waist. The eyebrows vanish completely, making the forehead and eye ridges appear larger and revealing a more prominent brow ridge. A small increase in muscle mass is evident, and muscle tone is sharply defined. Energy radiation is so great that the aura pulses at an extremely high frequency, almost to the point where it seems static; the sound of the aura is also pitched even higher than that of Super Saiyan 2. Bio-electricity, like in the Super Saiyan 2 form, is again constant, and may reach further outward from the body than before. Finally, the Saiyan's voice may become slightly deeper, though this is obviously a feature only found in the anime. There is no personality change, as no emotion is required to initiate the transformation (Goku acted quite stoic to intimidate Babidi and Buu, and Gotenks acted as he normally would). Another notable change is that the user seems to "glow" while in this state.
Goku first used this transformation in a brief battle against Majin Buu in order to stall him long enough for Trunks to find the Dragon Radar. He was able to overtake the monster, completely outmatching him blow for blow, though the obese version had yet to freely demonstrate his full power by getting angry at the time. He eventually retreated from the fight, as he wanted Goten and Trunks to defeat Buu with their fusion technique, as he "wouldn't be around forever." (He later states he may have been able to kill Buu if he really tried, though he nevertheless holds complete faith in Gotenks's ability to destroy him, if they were to try the Fusion Dance as Super Saiyans.)
This form was next seen used by Gotenks, the fusion between Goten and Trunks. While in the Hyperbolic Time Chamber, Piccolo destroys the only entrance in or out of the dimension, in hopes of trapping Super Buu inside, when he believes that Gotenks has lost and that the universe will be destroyed. Unfortunately, Buu was able to escape by ripping the dimensions open using intense sound waves from his own voice when he became angry at the thought of not having anymore sweets. Gotenks followed suit by transforming into a Super Saiyan 3 and ripping a hole in the dimensions in a similar fashion, allowing himself and Piccolo to escape. Once outside, Gotenks battled Super Buu in earnest, only to prematurely separate into Goten and Trunks before delivering the final blow after overwhelming him (the transformation had presumably drained Goten and Trunks' individual reserves as Gotenks, and because Goten and Trunks had differing power levels, the fusion destabilized).
In Dragonball Z Movie 12: Fusion Reborn, Goku used the Super Saiyan 3 transformation to defeat Janemba, only to have his foe transform into an even more powerful state. This new form tested the limits of Super Saiyan 3, and despite holding his own for a good while, Goku was eventually defeated in battle. It took the combined efforts of Goku and Vegeta, fused into Gogeta, to take down the demon.
The transformation was truly pushed to the breaking point in the final battle against Kid Buu. Despite using his full power as a Super Saiyan 3 from the very start (in the anime Goku started of as a Super Saiyan 2 before ascending), Goku had difficulty competing with the seemingly endless power supply of the Majin. Though he had an enormous advantage for most of the fight before the ki drain became evident, his attacks seemed to be useless against Buu, and the transformation eventually caused his collapse from exhaustion. He later reveals that, he may have been able to defeat Kid Buu when at full power. Vegeta then powers up to Super Saiyan 2 in order to buy him the time that he needed to get back up to full strength, but the immense stress of trying to gather his power forced him to revert to his normal state before reaching his maximum potential, leaving him heavily drained. It required a superpowered Spirit Bomb gathered from all the energies of Earth, the Ogres in Yemma's office, and the Namekian villagers that resided with Moori on New Namek, to finish the demon, and the form is never seen in the main manga storyline again.In Dragonball Z Movie 13: Wrath of the Dragon, Goku battled the evil monster Hirudegarn in his Super Saiyan 3 form. He was only able to kill the monster by using his Dragon Fist technique. Goten and Trunks also fused into Super Saiyan 3 Gotenks during the fight with Hirudegarn. He used his Continuous Die Die Missle attack on Hirudegarn's first stage, followed by a ki blast which seemed successful in destroying the monster. However, Hirudegarn was still alive and transformed into his second form and swatted Gotenks into the ground, hitting him so heavily that he defused and reverted back to Goten and Trunks.
In Dragonball GT, Goku's child form caused him to have trouble using Super Saiyan 3. His small body could not handle the energy output of the transformation, and he could only hold it for a few seconds. When he attempted to fight Baby Vegeta, he was easily beaten down without much effort. After having his tail restored, he becomes able to freely use the transformation, but the gap between Baby Vegeta's Super form achieved after receiving all the energies of the half-Saiyans and the Earthlings that he possessed was far too big for him to even land any damage, and he reverts to his normal state, only to transform into a Golden Great Ape, and then into Super Saiyan 4, defeating Baby after a prolonged fight. It was after this fight that Goku discontinued the usage of Super Saiyan 3, as it was heavily outclassed by the raw power and minimal setbacks of Super Saiyan 4.