There can't be, since KC isn't computable (could be mistaken on that in itself precluding a bounded error).
KC may not be uncomputable in general, but I'm pretty sure that doesn't preclude all possible proofs or constructions*, and it seems odd to say that there is no upper bounds when we have what sure look like such things.
* I have just invented a Scheme-like programming language in which all tokens except numbers are 2 alphanumeric characters long; what is the Kolmogorov complexity of the bitstring '1'? 'Who knows, KC is uncomputable -' Bzzt! The answer is that the KC is exactly 1, since the shortest program which emits the bitstring '1' is a program consisting of the constant '1' which evaluates to '1', which as you can see, is indeed of length 1, and all other programs emitting the bitstring '1' are longer by definition.
Or if you don't like that example, consider taking your favorite language and enumerating all possible programs in length order; consider the outputs of the programs you've just enumerated when evaluated up to the highest available Busy Beaver in steps (to avoid nontermination issues), and the respective lengths of outputs - have you not found the KC for those outputs? If you run gzip over the outputs to get estimated KCs, are those not upper bounds on the actual KCs you proved?
Computing upper bounds on on Kolmogorov Complexity is not very difficult: gzip and all the other compression algorithms do it. The difficulty is computing non-trivial lower bounds:
For all programming languages (with a self-terminating encoding), there is a trivial lower bound that doesn't depend on the string. This bound is at least one token.
But there is also a language-dependent constant L_max which is the maximum KC complexity and lower bound on KC complexity that you can compute for any string: L_max is the length of the shortest program for which the ...
I know this reeks of witch-hunting, but... I have a hunch that u/Eugine_Nier is back under the guise of u/Azathoth123. Reasons:
I don't have an axe to grind against the guy, I've only spoken to him a couple of times and didn't notice any particularly large karma hits afterwards, I just really dislike it when someone skirts the rules like that. Disruptive users evading permanent bans never helped any community ever.
Obviously I'm posting this here because I think a moderator should look into the matter. Usually I would be posting a disclaimer of some sort, apologizing in advance to Azathoth123 for attacking his standing with slanderous accusations if this turned out not to be the case. Well, I won't. The more I look into the matter, the more confident I get that they're the same person. Azathoth, if you're reading this and you're not Eugine_Nier, then I strongly advise you go search for your twin brother, I think you'll get along very well. Seriously, I'm saying this in good faith. You have a suspiciously great deal of things in common.
If retributive downvoting is (still) a concern (if not, then disregard this paragraph): I'd like to request, if such a thing is possible, that a mod karma-blocks me until the issue is over, so as to not incur undeserved downvotes (it would also mean I'd get no upvotes). In turn, I promise not to abuse the system by spamming the boards with garbage without consequences, but then again given my history so far on LW I don't think that such an abuse should be expected from me. For the record, I could have made a throwaway account just to say this, and not risk being karmassassinated, but 1) a zero karma account has no credibility and 2) for signalling reasons I prefer to put my money where my mouth is.
P.S. I only made this announcement its own post because the latest open thread was about to "expire".