In Dragon Ball Z: Lord Slug, after Piccolo transfers his energy to Goku, Goku uses the Kaio-ken x100 to rip a hole through the giant Lord Slug. In Dragon Ball Z: Cooler's Revenge, during Goku and Cooler's initial fight, Goku seemingly uses the Kaio-ken (though unnoticeable, as it was in "Bold and Fearless").
Ultra Instinct like the one you're talking about will never use Kaioken, because Goku's body is already at it's max potential when using it. If he were to add Kaioken, he would likely die immediately.
So far, Goku has only reached Kaio Ken times 20 on screen. However, Goku may be able to access Kaio Ken X 50, 100, or 1,000 with enough training! The toll that KK takes on its users is another reason most don't use it - you tend to be sore and tender after long periods of use.
Why doesn't Goku use kaioken more? Well, he kinda does, but he combines it with SS Blue in Super. He hadn't used it by itself because he gained all those Super Saiyan forms and Ultra Instinct so he didn't need it anymore.
Xeno Goku can use up to Kaioken x100 which normally would put strains on his body. Range: Extended melee range. Multiversal+ with ki blasts and attacks.
The highest multiplication for the Kaio-ken ever seen was the Kaio-ken x100, which Goku used against Lord Slug after absorbing a Spirit Bomb.
The highest multiplication used was X20. Both instances was against Freeza, which inevitably was the final time Kaio-ken was used in battle. Uniquely in Universe, Goku's maximum multiplication is just X10.
By using the power of Kaio-ken while in his Super Saiyan form, Goku gains a massive power increase, which is statistically equivalent to that of a Super Saiyan 2, by taking the 2x multiplier of these forms into consideration.
In Dragon Ball Z, Goku has the Kaio-ken as one of his special moves, and he is able to use it even in his Super Saiyan form. In Super Dragon Ball Z, Goku is able to use Super Kaio-ken by transforming into a Super Saiyan and then using Kaio-ken.
The introduction of Kale confirmed pretty much that the movies are not canon to Super at the very least, so for that series Goku never achieved Kaio Ken x100 or encountered Lord Slug to begin with.
Goku has been able to showcase the ability to lift over 100 tons with relative ease. He is even strong enough to lift a mountain over his head.
Kaioken x100 multiplies your level of power by 100, hence the '100x' beside the 'Kaioken'. If a fighter is able to handle it, they will be able to match a Super Saiyan 2, if their base power levels are somewhat equal.
One way to put it is that the False Super Saiyan transformation occurs when a Saiyan has the required physical power but lacks the required mental strife to transform into a Super Saiyan.
The power increase in Omen is far beyond that of Super Saiyan Blue: roughly equal to Kefla's Super Saiyan 2 state and allowing him to initially fight evenly with Jiren. Whis describes the power of this omen as the "limit-breaking power of the gods" (限界越えるネ申の力, genkai koeru kami no chikara).
Later on, because of the danger that using Kaio Ken poses, it would make sense for it to not be used. Goku or King Kai would probably not want the other Z Fighters putting themselves at such a huge risk for questionable payoff, so they wouldn't bother teaching the technique anymore.
Goku Black's body is post Tournament of Destroyers Goku's body, and he might have assumed that using Kaioken with Rose would destroy his ability to control ki like it did Goku for a while, and Zamasu was afraid he would never recover. After all, using SSB Kaioken X20 for Goku was a big step.
Jiren is a protagonist from Dragon Ball Super who is widely regarded as one of the most powerful fighters in the Dragon Ball universe. Many fans consider Jiren to be stronger than Goku based on the exceptional abilities and feats that he demonstrated in the Tournament of Power arc.
Kaioken (Namek Saga) has 20X multiplier to base, making Kaioken 2X stronger than Great Ape.
King Kai himself had stated that he couldn't handle the Kaio-ken. Despite developing the technique, he was never able to master it and so never used it himself (it would be cool to see him use it though).
Yes. There is a limit which bars Goku from using something like kaioken 100x or 1000x! It is the physical barrier which gets down when Goku achives a whole new level of power. The kaioken can kill Goku if he uses it for longer duration or at a level of greater amplification of power.
There's no such thing as a Kaio-ken x1. There can't be. The Kaio-ken is a technique in which the user's fighting abilities are multiplied by a stated amount, Kaioken x2 doubles, Kaioken x 3 triples, Kaio-ken x4 quadruples, Kaio-ken x10 decuples (there's a word I bet you didn't know), etc.
Anyone who managed to master the Kaio-ken would automatically be stronger than anyone who has not mastered the Kaio-ken. Though, the Kaio-ken is already theoretically infinite, it just has greater and greater strain based on the level of the attack.
Dragon Ball Super not only took time to introduce the new god like transformation in Ultra Instinct, it allowed Son Goku to utilize Kaioken in an entirely new way by accessing it while in his Super Saiyan Blue state and the franchise has recently dropped two new bombshells when it comes to these iconic power boosts!