Q:

Grammar framework features that are not supported cross-linguistically

There are quite a lot of grammar frameworks postulated since the last century, like MP, LFG, RRG, RCG, MTT to name a few. I like reading about languages, but a lot of publications about languages are couched along such grammar frameworks, and for someone like me who has no academic background in Linguistics and has not read a lot of  linguistics literature, its difficult to evaluate which grammar framework has more explanatory power, or which of their features are robust cross-linguistically.
It's probably contentious to ask which is the best, so what I would ask instead is, if we could come up with one particular feature, characteristic or attribute in a specific grammar framework that is invalidated cross-linguistically and which grammar framework addresses it best. It would be ideal if in the reply we could specify which languages and language features are counter-evidences of which grammar framework features. This is to avoid closing this question for not being constructive (soliciting "debate, arguments, polling, or extended discussion"). I'm thinking along the lines of responses like this and this. 
As much as possible, please limit each answer to one feature per grammar framework, unless a set of grammar frameworks share this feature.
Also, if its possible, can this be made a community wiki? 
As an example, this excellent discussion about binary branching of all syntactic structure in MP is a very good example and is the one that made me think of this question.

EDIT: In light of @prash's comment, what I am looking for are falsifiable "predictions" that flow out of these grammar frameworks/formalisms, in a Popperian sense. What cross-linguistic data could prove a formalism's feature to be true or false as against other formalism? Instead of looking for data that support such formalism, one should look for data that falsifies such. In the binary branching of all syntactic structure discussion, MP posits a binary nature of syntactic structures which Tim Osborne show is untrue. The weight always goes to nonconforming data. 
Even if all formalism can accommodate all cross-linguistic data, applying Occam's razor can tell us which formalisms have the same explanatory power in spite of fewer assumptions. 
That by itself is a count against some formalisms. Actually, I don't know much about these things, I was just following the discussions and very much interested in knowing other features which fall under the same category.

A:

I agree with the premise of the question that the first task of a scientist is to devise a framework that makes potentially false predictions, and thus that looking for false predictions is good scientific practice. It is important to note though, that this game is better played towards one's own favored scientific theory, if only because this game is embarrassingly easy to play already for the most advanced of the exact sciences (just ask a physicist what dark matter and dark energy, which are supposed to constitute some 96% of the Universe, are supposed to be) and so becomes almost inane for a field in its infancy, like linguistics (anyone with a theoretical framework in linguistics should be able to produce on demand ten phenomena the framework cannot explain at all at present, otherwise the framework is simply not serious). Criticisms of other theoretical frameworks can be valuable, but should be done by proposing competing analyses, not by pointing out defects in actually existing ones (so to continue the analogy, by explaining the motions of galaxies and galactic clusters if one does not believe in dark matter). With that in mind...
Core assumptions of minimalist syntax
Usual core assumptions and goals in minimalist syntax are as follows. 
1) There exists an early stage of syntactic computations (narrow syntax) the objects of which are constructed by repeated and recursive applications of a binary operation (Merge) in a bottom-up fashion. 
2) The only operation beyond Merge allowed on these narrow syntactic objects is the operation Agree (that is features co-valuation) and both Merge and Agree are subject to strict locality conditions.
3) There exists specified and definite domains of computation (phases) the completion of which triggers transfer of the narrow-syntactic object somewhere else (the interfaces) with no return possible.
4) Everything else is interface conditions.  
Potentially false predictions following from these assumptions
There are tons of them, because the assumptions above are extremely counter-intuitive, in fact worryingly so. Here are a few examples.

Sensitivity to hierarchical rather than to linear order.

It follows from 1) that narrow syntactic objects are necessarily unordered binary trees. Hence, minimalist syntax predicts that the relevant structures for syntax are those prevailing in binary trees (for instance C-command) and not the most obvious linear ordering of the words in the sentence.
With the caveat that no language wears its analysis on its sleeve, if one could find a language for which, say, negation was licensed by a phenomenon taking place at the third word of the sentence, or in which adverbs modify the closest verb in the linear order rather than the closest one in the hierarchical order (the infamous Instinctively, eagles that fly can swim example of Chomsky), or in which cataphoric pronouns were distinct from anaphoric pronouns, or in which questions words were in situ with no intervention effects, or in which words were systematically interpreted where they are pronounced, or for which binding of reflexives depended on linear precedence, or if there existed a V3 language in the same sense that there are V2 languages, then minimalist syntax would be dealt a serious blow.

Successive cyclic movement

Because narrow syntactic objects are constructed from the bottom-up (extremely weird), words appearing at the left-periphery of the sentence sometimes (often) have to be merged far deep in the hierarchical structure. Because computations is done by chunks, they then have to move at least out of each chunk to reach their final destination (weird). And finally, because the only way for them to move is to being remerged (extremely weird), the potentially false (in fact prima facie seemingly absurd) conclusion that sentences are riddled with copies of words all the way from their departure point to their final stop ensues.
So the existence of a language immune to island effects, or parasitic gap effects, or obligatory versus optional control effect, or without reconstruction effect would deal a serious blow to minimalist syntax. Beside, minimalist syntax would not have lasted 5 minutes (in fact would not have been born) if said copies were not occasionally found in some languages and constructions precisely where they were hypothesized to be.

Independence from the interfaces.

This is obvious, but well-worth being pointed out nevertheless. Minimalist assumptions affirm the independence of syntax from the interfaces. If there existed a language whose syntactic rules depended on phonology or for which, for instance, island effects would show up only for meaningless or unclear questions, then minimalist syntax would disappear. 
A less trivial potentially false (in fact, again, prima facie absurd) conclusion occurs when this principle is taken conjointly with morphology: as soon as one attributes some overt morphology phenomenon to hierarchical position in minimalist syntax, then one has to assume that all languages exhibit (possibly abstractly) the same morphology effect in the same hierarchical position.This would be proved false, and put minimalist syntax in serious trouble, if for instance a language was found in which licensing of DP by nouns and by verbs were identical. 

