Learn English – How did the word “chunter” come about

Umpteenth comes from umpty, meaning an indefinite number. Etymology Online says "umpty" is derived from "Morse code slang for "dash," influenced by association with numerals such as twenty, thirty, etc."

And I think the "teenth" insert is to make the "th" easier to pronounce for the ordinal formation, but it might be also to add emphasis by way of lengthening the word. Also, it might be because of the teen/ten connection I mentioned in my comment above: an indefinite number of tens.

And this source suggests that Morse code slang word "ump" is imitative in origin, so I gather that means that that's what a dash sounds like in Morse code.

It occurred to me that this means something like _th times, then, or the equivalent of nth times (to the nth degree?)—although a single dash is Morse code for T.

According to the OED Online, "re" is an English preposition arising from a Latin borrowing,¹ meaning

Originally: in the matter of, referring to; in re (see in prep.2 23d). Subsequently: about, concerning.

1707 T. Hearne Remarks & Coll. 17 May (O.H.S.) II. 14 Amused by Charlett's trick re Tacitus.

("re, prep.". OED Online. June 2016. Oxford University Press.)²

Thus re has been a word since ancient Roman times (as your own definition shows), and has been in use in English since at least the early 18th century.

A related question might be:

Why do we use re the way we do, and where did the perception that it is an abbreviation originate?

The OED notes that

The use as a preposition was formerly much criticized in usage guides. See e.g. H. W. Fowler Mod. Eng. Usage (1926) 255/2, and compare quot. 1935, which parodies this use.

The form re. probably results from reanalysis as showing an abbreviation for regarding prep.

So our current usage dates from at least the early part of the 20th century, and the abbreviation version probably arose from a folk etymology.

A fuller explanation might be hinted at in the reference in the definition to in re. This is a legal Latin term, where it's used to refer to cases that don't have two sides:

Jones v. Smith = Jones and Smith are fighting it out in court.

In re Jones = the court is doing something about Jones, but there's only one side (e.g. an adoption).

By extension, it also has long been used in the legal profession when referring to any case in legal memos, so

In re Jones v. Smith = about the Jones against Smith case

And by further extension, to refer to the subject matter of a memo in general:

In re(:) overdue payment

This memo usage also appears in business; I suspect that the business usage arose from the legal usage, but haven't been able to confirm that. Latin has been a language of English law for so long that, while there are numerous lists of legal Latin phrases on the web, the only discussions I'm finding of the history of this use are in places like a "For Dummies" website and some Quora discussions. I assume that Latin has been a staple of English courts since the Romans occupied Britain, however, and it's plausible that business folks would have needed to be familiar with common legal terms.

According to the OED Online, the confusion of the Latin preposition "in with the English preposition "in" is of long-standing, as evidenced by the fact that in some early examples it is "found printed in roman type, while the rest of the phrase [whatever The Latin phrase may be] is in italics." ("in, prep.²". OED Online. June 2016. Oxford University Press.)

With an ambiguous "in" attached to such a short little word as re, and meaning in context "in regard to" or "in reference to", it's not surprising that the Latin phrase in re or just re has frequently been taken to be an abbreviation of an English phrase.

Indeed, at this stage, I believe "Re:" actually is an English abbreviation, in the sense that many (most?) writers of English who use it at the top of a memo or email mean it to stand in for some English word, whether "regarding" or "referencing".

However, its use as a standalone word is also well-established and "correct".

1. From the OED definition cited above: "Etymology: < classical Latin rē, ablative of rēs thing, affair (see res n.¹)."

2. As "re" is in the ablative case, I think the translation would be something like "from (because of) the thing" (some Latin scholar please confirm or correct).