grammar - Is it possible for a word in a sentence to have two grammatical functions at once?

I just saw this question, which is about the whoever/whomever choice in these sentences:

I will kill whomever I despise.
I will kill whoever despises me.

It made me think; what is the object of the verb kill? Is it not who(m)ever? Then what is the object (subject) of the verb despise in the first (second) sentence?

And in general, is it possible that a word should have more than one grammatical function in a sentence? How?

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Edit:
_{(This edit is a about the specific examples given above. If that's not an issue for you, just ignore it and read Greg's post, which answers the question in the title.)}

Turned out my examples were not to the point. I Quote from A Student's Introduction to English grammar by Huddleston and Pullum, p.191:

Fused relatives
The final relative construction we consider in this chapter is the fused
relative, illustrated in [19]:

• [19 i]: Whoever said that was trying to mislead you.


• [19 ii]: I 've eaten what you gave me.

This is a more complex construction than those dealt with above. Here the
antecedent and the relativised element are fused together instead of being expressed separately as in simpler constructions. The underlined expressions here are thus NPs whose head is fused with the first element in the relative clause.

Whoever in [19 i] is simultaneously head of the NP and subject of the relative
clause, and its gender indicates that we are talking about some person. The meaning is thus comparable to that of the non-fused

• The person who said that was trying to mislead you.

What in [19 ii] is likewise head of the NP and object of gave in the relative clause, and the non-personal gender gives a meaning like that of the non-fused (and more formal)

• I 've eaten that which you gave me.

So in I will kill whoever despises me, the object of kill is the whole relative clause (i.e., whoever despises me), and whoever is simultaneously the subject of despises and the head of the clause.

I'm not really comfortable with the idea of an NP whose head is a dependent of an element (here, the subject of a verb) inside that NP. It's a loop. There may be another way to describe this construction, so this loop could be avoided, but I'm going to leave it here for now.

Answer

In grammatical theory, for theories based on hierarchical tree structure, No, because presumably grammatical functions correspond to the descending lines of a tree diagram, and no two lines can converge, allowing a single item to be the daughter of two different mothers. I know of two theories which do not require strictly hierarchical trees and which do allow a single item to have two different functions.

One is Relational Grammar (and Arc Pair Grammar), proposed by Paul Postal, David Perlmutter, and others. However, although it is proposed in this theory that some tree branches converge, I don't know that evidence has been given that this is actually so.

The other is McCawley's variety of transformational grammar, described in his textbook The Syntactic Phenomena of English, which includes an explicit account of a modified phrase structure theory that sanctions converging (and crossing) tree branches. Specifically, McCawley proposes that the raised node of right-node-raising (RNR) constructions simultaneously has grammatical functions in both of the conjuncts to its left. So, for instance, in

John built, and I installed the stove of, the new kitchen.

the raised constituent "the new kitchen" is simultaneously the direct object of "built" and object of the preposition "of".

McCawley does give some evidence for his theory of RNR constructions that is based on how the CNPC constraint works in some rather complicated examples, which I cannot recall in detail.