I've said it once, I've said it twice, I've said it a thousand times

Solution 1:

The reason English has three different words for those is because English has three different words for 1, 2, and 3. It’s like why we have three different words for sixth, eighth, and twelfth: there’s a suffix here used with regular numbers.

The difference is that instead of ‑th for ordinals, it’s ‑ce for adverbials, and you just aren't recognizing that ‑ce adverbs things — or at least, that it adverbed them once upon a time.

That’s because ‑es was a genitive adverbial suffix in Old English. You can see its remains in all kinds of adverbs that are today spelled with an ‑s at the end.

Note that genitive nouns also end in ‑s. The difference is that making genitive nouns (well, and noun phrases with a clitic ‑’s) is still productive. However, making adverbials this way no longer is so, although now and then people coin new ones along existing models by analogy.

Other ‑s examples of adverbs made like once, twice, thrice include such words as afterwards, backwards, besides, betimes, forwards, hereabouts, needs, nowheres, nights, nowadays, sideways, thereabouts, towards, unawares — with plenty more where those came from. As Janus mentions in comments, there is also a broad set of those which gained a parasitic (=inorganic, non-etymological) final ‑t on top of their ‑es/‑s, such as acrost, against, amidst, amongst, betwixt, gainst, whilst.

Because we made adverbs out of things by adding ‑es (later ‑s) all the time, it was the natural way to make an adverbial out of the numbers. The word nonce has the same origin, although that’s used for a noun not an adverb. In Old English, the word for modern once was ænes or enes, genitive forms coming directly from the Old English word for one, which was án.

By the way, the Old English word án that gave birth to modern one and once also gave us the indefinite articles a, an. But when Old English speakers said ænes (again, that was their word for once), it had two different syllables. That situation was not to last, however, and this led to orthographic changes.

Middle English picked up twice and thrice by the same construction, although with differing spelling and pronunciation than we use today. Then around 1500, as ones became monosyllabic, it began to be spelled -ce to indicate the lack of voicing — and, for the cases of twice and thrice, to indicate a change in character of the preceding vowel.

So what’s going on is that you no longer recognize this fossilized morpheme as meaningful. It means exactly the same thing as those other free morphemes you mention in other tongues.

And no: twice isn't going away. Neither are twin, both, halve, or double. Twain might someday be, though, moving into literary use not bar talk. As for your Italian learners, I strongly advise against saying “two times”; it does not sound right to this native ear. Indeed, the OED specifically reports that twice is:

In all senses now the regular substitute for the phrase two times

On the other hand, tuppence and thruppence are hardly worth a farthing today, eh? :-)

Solution 2:

So why does the English language have three different words for "one time", "two times" and "three times"?

In other words, why do one time, two times and three times have single words (once, twice, thrice) but four times, five times etc. don't have?

Simple answer is; one time, two times and three times were frequently used—as lower numbers like one , two and three tend to be used more frequently—and turned into one word naturally.

From a linguistic standpoint, the adverbial use of the genitive of one was lexicalized and extended to two and three. The morphological formation involved in this process is fusion.

The usual word for this sense of times in Old English and early Middle English is sithe. (probably a softer pronunciation of tide)

_{In Old English the case is either the instrumental, or the accusative governed by a prep. The instr. pl. síðum became Middle English sīðen, and finally assumed the same form as the sing. In place of a numeral, an adj. or adv. might be used, as in eft-sith(es) eft adv. Compounds, fele-sithe(s) fele adj., and oftsithes adv.). With the Scottish forms (δ) cf. modern North Frisian -sis, as in twasis twice, manningsis many times.}

For example, OED lists the earliest phrases as æne siða and ænne sið for one time:

• c825 Vesp. Psalter lxi. 12 Æne siða spreocende wes God.
  
• OE Beowulf 1579 Oftor micle ðonne on ænne sið.

Then, we can see that it starts forming into a single word (ænes, anes, ones) and eventually becomes what we call once today:

• OE Lambeth Psalter lxxxviii. 36 Semel iuraui in sancto meo : ænes ic swor on minum halgan.
  
• ?c1200 Ormulum (Burchfield transcript) l. 1078 Þatt wass aȝȝ æness o þe ȝer.
  
• a1225 (▸?a1200) MS Trin. Cambr. in R. Morris Old Eng. Homilies (1873) 2nd Ser. 109 (MED), The sunne..arist anes a dai.
  
• a1325 (▸c1250) Gen. & Exod. (1968) l. 3288 Ilke dai..Ones he ðor it sungen rigt.

But then, why it is changing back to two separate words (thrice > three times)? This is related to the typological shift in the language from synthetic to analytic by language historians. [Synthetic languages have a high morpheme-per-word ratio and analytic languages have a low morpheme-per-ratio]

Sources:

• OED
• Naturalness and Markedness by Henning Andersen / academia.edu
• Naturalness and Iconicity in Language edited by Klaas Willems, Ludovic De Cuypere
• Wikipedia - Lexicalization, Synthetic language, Distributed Morphology

Beyond a shadow of a doubt the English thrice is doomed to exile if not extinction, is twice suffering the same fate? For instance, many Italian learners do say "two times"—for them it makes more sense.

Language is changing but morphological changes or formations slowed down in Modern English. Twice might seem like changing into two times but it still sounds more natural in most contexts and it is more common; however two times is gaining usage in informal speech mainly.

Twice wins the usual Google Ngram battle obviously but there is, of course, more to it. Below is an analysis and comparison of twice and two times based on selected British and American newspapers (from the book One Language, Two Grammars?: Differences Between British and American English edited by Günter Rohdenburg, Julia Schlüter).

While once is firmly established and thrice has been generally ousted by the more regular (analytic) three times, there seems to be a competition between still-frequent (synthetic) adverb twice and two times. ^(rephrased)

The data in the figure show the rate of occurrence of two items in certain high-frequency collocations.⁷ Though twice is still well entrenched here, it is used more sparingly in AmE than BrE. The frequencies of two times contract in the reverse direction. This suggests that there might be a compensatory relationship between the two adverbs, with AmE favoring the more regular option.⁸

_{⁷ The environments searched includes twice/two times as much/often/large, etc. twice/two times the size/length/speed, etc. and twice/two times a day/week/year, etc.
⁸ The case of special temporal adverb twice has a (distant) parallel in the time expression fortnight and the derived adjective/adverb fortnightly. Here again, even the formal AmE makes much less use of the synthetic and more opaque term: in The Washington Times,forthnight(ly) occurs merely 1:5 times pmw, while in the British Times it has a frequency of 28.4 pmw.}

OED is bold enough to say the below but it might not be considering all the relevant data:

two times as advb. phr. (expressing repetition or multiplication) is now used only with a demonstrative or defining word; otherwise twice is substituted

• 1574 E. Hellowes tr. A. de Guevara Familiar Epist. 174 Two times I haue moued the Cardinall Tortosa in your busines.
  
• 1916 N.E.D. at Two, Mod. I have known it happen two separate times. I called upon him three times, but saw him only once; the other two times he was away.

Solution 3:

Jim Morisson would have sounded funny singing 'Love me twice baby, love twice today, love me twice girl ..."

But then, Lennon and McCartney's "One after 909" works better with "Move over once, move over twice, c'mon baby don't you be cold as ice," because "two times" does not rhyme with "ice."

But while we are asked to "Knock three times" on the ceiling, we should knock "twice on the pipes" if the answer is no.

It's a matter of style, rather than correct grammar. While brevity is desirable, the lengthier "two times" has its place if it serves to emphasize or enhance the cadence.

And as an example of when one should say "two times" rather than "twice", we would say "The last two times ..." rather than "The last twice ..."

Solution 4:

(1) Taio Cruz - Dynamite

(Pop song lyrics are a great pedagogical device in these cases. Link)

The idea that twice is facing extinction seems whacky, look up the plays of that song.

Note that the pop song industry targets the lowest conceivable intelligence and educational demographic market...and indeed targets LCD international English.

(2) "twice" is a commonplace word. All you have to tell the students is "that is a commonplace word."

(3) "However on reflection, his question was a valid one." No, it's silly. All he or she can possibly do is ask some English speakers if it is a common word. If they all say "sure", then, if he/she disagrees it is bizarre.

(4) Regarding Latin, German etc - the issue would seem to be irrelevant, right? I can see no connection, at all, between etymology and "what words are currently commonplace".

We could trivially list a number of words that are currently very popular (with such and such etymology), and we could trivially list a number of words that are currently not popular (with such and such etymology). Right?

(5) Hence regarding your question... "Is 'twice' heading to extinction?"

As far as I know the only way to answer such questions is (a) demonstrate the word is used in commercial hugely successful pop music industry lyrics or (b) just canvas opinion of native speakers. I'd say "sure, it's commonplace".

For example, parents often yell at the kids "don't make me ask you twice, damnit!!" It would be very unusual to yell "...two times" "think twice" "i had to run to the office twice today" are all highly natural, common phrases (with the usual caveat that any such claim about "what is common" in language is just an opinion).