As far as I know, no such limit has ever been stated, but to date Goku has never used a multiplier higher than 20.
I suppose the argument can be made, then, that x20 is just the technical limit of the Kaioken technique. In theory, then, Goku, instead of acquiring the Super Saiyan transformation, could have simply taken his Kaioken technique as far as x50 without suffering any noticeable drawbacks for it. As I mentioned earlier, if they were really all that different, then there's no reason why Goku couldn't have stacked Kaioken on top of Super Saiyan, at least as early as the Cell saga, except to consider that perhaps his body could not actually withstand such a vastly greater multiplier on top of what Super Saiyan already afforded him. I theorize that since Goku showed no signs of wear and tear from using Kaioken x20 on Namek when he first faced Freeza, and since he soon after achieved a transformation that pushed his body to 50 times it's previous limits, then, in my mind, there must be little to no actual differences between the form and the technique's toll on the body, and Goku's body had simply reached a level that it could endure being pushed as far as 50 times it's current limit. A potential Kaioken x50 should have the exact same power output effect as the Super Saiyan transformation, right? Would it exact a higher toll on the user's body, mind, spirit? If so, then why?
They make you a better, stronger, faster warrior through large, precise leaps. My thought, then, was, "what makes the Super Saiyan transformation and Kaioken different?" They both simply multiply one's power level, that's that. Kaioken on the other hand, while still not exactly living up to its potentially lethal warning label, seems to carry more narrative weight as a risky, draining technique that has to be used sparingly. Now, since it's inception, Super Saiyan has been retconned to have more of a stamina cost to it than fighting in base form, but, that's really only ever been stated hasn't it? I can't recall a single time when a fighter using Super Saiyan form has ever exhausted themselves to the point that they just couldn't fight anymore, at least not without also being first beaten within an inch of their lives, which kind of takes you out of the fight regardless.
So, obviously a 50 times power level multiplier from Super Saiyan with little to no discernible drawbacks is dramatically more beneficial than a 2x, 4x, 10x, or even the 20x multiplier offered by Kaioken since Kaioken was introduced as something that is incredibly strenuous on the body and potentially lethal (even though Goku, the only character ever seen using it, has not one time ever been harmed by it.
Goku and all of the other saiyans have this transformation that just multiplies their power levels with little to no drawbacks (though differing forms certainly vary). Goku used Kaioken before he achieved the Super Saiyan transformation, but after becoming a Super Saiyan he never used it again until he achieved the Super Saiyan Blue transformation, which is basically described as a perfect Ki control Super Saiyan God Super Saiyan transformation, a Super Saiyan with the power of a God if you will.