During the FeforGrammarians discussion on immediate challenges to developing our grammars (FeforImmediateChallenges) we indentified a set of phenomena which seem to require access to indices other than the XARG belonging to saturated arugments. These include:
Floated quantifiers (including floating numeral classifier phrases in CJK)
Various languages of the Americas in which verbal arguments seem to be saturated by proniminal affixes, where independent NPs appear to be modifiers further specifying those arguments.
Secondary predicates in English ("Kim hired Sandy drunk") and Norweigan ("Inkvisitoren torturerte presten andektig" - 'The inquisitor tortured the priest in devotion'.). In both cases there is evidence that the secondary predicate attaches to VP (not to the object NP). In English, the string is ambiguous, with "drunk" being predicated of either Kim or Sandy (? but not both). The same is true for the Norwegian example given above, meaning that either the inquisitor or the priest is in a devoted state. The example can also mean that the whole act was done in a devoted manner. For many such cases in Norwegian, the morphology on the secondary predicate disambiguates. It forces the low (semantic) attachment in a case like 'Per drakk ølet kaldt' ('Per drank the beer cold'), where 'øl' is neuter and 'Per' masculine, and 'kaldt' has the neuter form; it forces the high attachment in 'Per drakk ølet kald' (which means that Per was cold). 'Per drakk ølet kaldt' can also mean that he drank the beer in a cold fashion, since '-t' is also an adverbial affix. In the latter case, ARG1 of 'cold' is the event index. In the subject predicative reading, it is identical to the verb's XARG, whereas in the object predicative reading, the ARG1 of 'cold' is equated to 'VARG', which exposes the object argument in the HOOK.
Let me add one more point to the list:
For some grammarians it might in addition be of interest to express the 'mover' as the ARG1 of the prepositional modifer in a directional construction, such that it is Per rather than the event that ends up in a park for the sentence 'Per runs to the park'. In NorSource, which has adopted this Jackendovian view, this has been implemented through the use of DIRCT-ARG parallel to XARG (and VARG).