Answer by Thomas Wier, assistant professor of linguistics at the Free University of Tbilisi:
Short Answer: Yes, but not in English.
What do we mean by valency, anyway?
To answer this question in more detail, it's important to distinguish different kinds of grammatical valency. As in chemistry, grammatical valency is a measure of a kind of asymmetry between a nucleus and potentially several satellite words that are structurally dependent on that nucleus. Contrary to popular understanding, almost any part of speech can serve as a nucleus around which other words will (optionally or obligatorily) surface: verbs, nouns, prepositions, adjectives, etc. The most common part of speech to show these effects is of course verbs, and verbs also have the largest variation in the kinds of other parts of speech that act as a satellite. Typically, when we talk about transitivity, then we are talking about how many noun arguments a verb takes:
Intransitive verb (V + N): John ages slowly.
Monotransitive verb (V + N + N): John kicked the ball.
Ditransitive verb (V + N + N + N): John gave Mary a book.
However, identifying discrete categories of verbs based on transitivity is not always easy, because (in English, anyway) there exist a number of verbs that seem to fall between the cracks, such as labile verbs and ambitransitive verbs. Labile verbs are verbs that may optionally allow a certain number of arguments:
John ate pizza—in fact, he ate all day long.
Here, in both clauses the semantics are unchanged, but in the first the semantic argument—the thing actually being eaten—is made explicit, while in the latter it is not. Like labile verbs, ambitransitive verbs also optionally allow varying numbers of arguments, but unlike labile verbs, the way in which the semantics is encoded as subject or object is different:
The cup broke—actually, John broke the cup.
Here, what is the subject of one clause is the object of the other. What such examples illustrate is that the semantics of a verb may remain constant—there is a breaker and a thing being broken in both cases—but the syntax of the verb may change. Syntactic encoding, in other words, is autonomous from semantic meaning.
What about tritransitive verbs?
So, here is where the debate about tritransitive verbs comes in. If we're using the number of syntactic noun phrases, as opposed to semantic arguments or other kinds of phrases, as our criterion, then English has no tritransitive verbs. All the other answerers for this question have provided examples that are either not obligatory (and therefore syntactic adjuncts) or are not syntactic noun phrases—or both. For example:
1) John traded Jane an apple * (for an orange).
2) I bet you two dollars (that it will rain).
Sentence No. 1 actually is merely a ditransitive verb with a prepositional phrase that, though semantically obligatory and syntactically requires a prepositional phrase headed by for, is not a noun phrase. Sentence No. 2 likewise is ditransitive with an optional clausal adjunct: If you remove the subordinate clause, the sentence is still grammatical, and can even be replaced with an entirely different and equally optional subordinate clause: I bet you two bucks, (because there's no way the Cubs will win the World Series this year). This distinction between syntactic transitivity and semantic transitivity is even clearer with certain verbs where the semantically obligatorily argument belongs to an open class. For example:
John put the book * (on the shelf/in his suitcase/beside the coffee table/down/away).
In this sentence, the location of the book is obligatory—*"John put the book" by itself is ungrammatical—but the exact kind of location is not specified by put. As long as there is some location specified there, whether that be a prepositional phrase or a locative adverb, the verb is satisfied.
Do strict tritransitive verbs with only noun phrases exist in any language?
Yes, they do, though they are somewhat rare. Most examples that on first appearance look like tritransitive verbs with only noun phrase actually have one noun phrase that is optional, e.g. French give:
Je te le (lui) donne.
I give you it him give.1Sg
"I give it to you for him."
In this example, the noun phrase lui (for him) can be optionally added or dropped as needed, and probably more often than not is not present. We might call this a tritransitive labile verb. Like English eat, it may or may not have that added argument. Note that this is not an example of differing semantic transitivity: The English phrase would have an identical meaning and number of arguments, but nonetheless * I gave you it him is ungrammatical in English.
There are some stronger cases of tritransitive verbs, and I will give three examples from languages of the Caucasus. In the Abkhaz language, for example, if the base form of the verb is already ditransitive, then an additional argument can be added with either the causative or benefactive valence suffixes (Chirikba 2003:39):
"I shall make you speak with him about it for her."
Unlike French or English, Abkhaz is a so-called pro-drop language: Because the verb is marked for agreement with every argument, no noun phrases are obligatory for any verb. However, this example still counts as a tritransitive verb because full noun phrases could be supplied if needed, and more importantly the verb morphology is not optional, if one wants to indicate that many arguments.
In another language of the Caucasus, Georgian, verbs can likewise indicate four noun phrase arguments at once, despite the fact that the relationship between inflectional morphology and syntactic arguments is not as straightforward as Abkhaz:
მე თქვენ ივანეს წიგნს დაგაცემინებთ
me tkven Ivane-s c'ign-s da-g-a-c-em-in-eb-t
1Sg 2pl John-DAT book-DAT PVB-2-PRV-give-TH-CAUS-TH
"I make y'all give John the book."
One last example is Svan, a Kartvelian language of the Caucasus rather distantly related to Georgian. In this language, there are double causatives that carry with it the implication of assistance in an act (Boeder 2003:43):
მǝშკიდ ხაშკა̈ა̈დუნე ჭყინტს ჩა̈ა̈ჟს
mǝšk'id x-a-šk'ääd-un-e č'q'int'-s čääž-s
smith.NOM 3-PRV-forge-CAUS-PRES boy-DAT horse-DAT
"The smith makes the boy shoe the horse."
ჭყინტ ხაშკა̈ა̈დუნა̈უნე მǝშკიდს ჩა̈ა̈ჟს
č'q'int' x-a-šk'ääd-un-äwn-e mǝšk'id-s čääž-s
"The boy helps the smith shoe the horse."
In this example, it is unclear exactly how many arguments are involved. It is clear that there are at least three. However, because the assistive causative example includes two causative suffixes, each apparently adding something to the sentence, it is not clear whether there is a fourth (in this case) implied argument meaning something like "The boy helps (the situation) that the man shoe the horse."
In any event, the bottom line is that tritransitive verbs with only noun phrase arguments exist but are rare.
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