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 technique, however, has its substantial flaws.
Mastery & Multiplication
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.
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").
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.
Potentially, Kaio Ken could be trained to go higher than 20x, there has been no official word stating either way. If there's a way to make it more powerful than Mastered Ultra Instinct (or a future transformation) or to use in conjunction with it, then we'll see it in the future of the show.
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.
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.
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.
The same power and speed, but the Ssj transformation alters the body to allow for the stress and strain of the increased power, while the Kaiou-ken increases the power, but doesn't adjust the body to compensate. Instant Death at that.
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.
Dragon Ball Super saw the Saiyan martial artist attain his most esoterically powerful form yet: the Ultra Instinct state. First unveiled during the anime series' final story arc, "The Tournament of Power," the transformation is apparently not a conscious one.
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).
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.
Reason being, Goku went Super Saiyan Kaioken in the Babidi saga. That was definitely anime only and was non-canon, and should be considered non-canon.
Of course he can, there is nothing that says he can't. Kaioken can be obtained by anyone.
We know Kaioken was invented by King Kai and goku is the only person to learn/use it. However Jiren is in an entirely different universe so it's possible he learned and honed his own kaioken. It would explain how he's able to attack and move so fast, plus it explains the red aura he displays.
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).
In the anime, since Ultra Instinct is a technique of the Gods, for mortals without sufficient divine training, there is one major risk of using Ultra Instinct.
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.
Super Saiyan 3 - Xeno Goku is capable of transforming into a Super Saiyan 3. This form is eight times stronger than an ordinary Super Saiyan. He uses this form to knocks out Gravy with a single headbutt in one of the games openings. Super Saiyan 4 - Xeno Goku is capable of transforming into a Super Saiyan 4 at will.
Xeno goku is basically got goku, but he meets all the gods and gains God ki then, unlike dbs goku who gets it during the ten year time skip between kid buu death and him vs uub. And yes, he's already stronger.
There is 95% chance of Ultra Instinct winning, but if somehow Ultra Ego user too has a good battle IQ, than ultra ego is not that far apart ultra instinct, so ultra ego might have the 5% chance to win. However, the creators have the choice to show who is actually better, because of course they are the creators.
So, if Ultra Instict Goku and Ultra Ego Vegeta fuses together into Gogeta or Vegito, they will combine two of the strongest techniques in the multiverse, increasing their possibility of taking down the God of Destruction Beerus.
Although Vegeta was taught the concept of Ultra Ego by Beerus, the God of Destruction of Universe 7, the transformation itself is unique to him. Characters such as Beerus can make use of this ability, but Vegeta is the only one who has access to the transformation of Ultra Ego.