What part of speech is “to” when used in infinitives?

Or is this a meaningless question that ascribes more importance to categories than to essence?

Since nobody else has taken up the cudgel, I’d say, judging from the OED excerpt below, that it’s a preposition.  This passage gives the history of the modern English infinitive with “to”.

History:Beside the simple infinitive, or verbal substantive in -an (ME. -en, -e), OE., like the other WGer. languages, had a dat. form of the same or a closely-related n., which in OE. ended in -anne, -enne, in ME. reduced successively to -ene, -en, -e, and was thus at length levelled with the simple infinitive, and with it reduced to the uninflected verb-stem. This dative form was always preceded or ‘governed’ by the preposition tó ‘to’. By many German writers it is called the ‘gerund’, after the Latin verbal n. in -ndum. In mod.Eng. the functions of the Latin gerund are more properly discharged by the vbl. n. in -ing, and it is therefore more convenient to speak of the OE. form in -anne as the ‘dative infinitive’ or ‘infinitive with to’. Originally, to before the dative infinitive had the same meaning and use as before ordinary substantives, i.e. it expressed motion, direction, inclination, purpose, etc., toward the act or condition expressed by the infinitive; as in ‘he came to help (i.e. to the help of) his friends’, ‘he went to stay there’, ‘he prepared to depart (i.e. for departure)’, ‘it tends to melt’, ‘he proceeded to speak’, ‘looking to receive something’. But in process of time this obvious sense of the prep. became weakened and generalized, so that tó became at last the ordinary link expressing any prepositional relation in which an infinitive stands to a preceding verb, adjective, or substantive. Sometimes the relation was so vague as scarcely to differ from that between a transitive verb and its object. This was esp. so when the vb. was construed both transitively and intransitively. There were several verbs in OE. in this position, such as onginnan to begin, ondrdan to dread, bebéodan to bid, order, bewerian to forbid, prevent, elíefan to believe, encean to think, etc.; these are found construed either with the simple (accusative) infinitive, or with tó and the dative infinitive. There was also a special idiomatic use (sense 13a) of the infinitive with tó as an indirect nominative, where logically the simple infinitive might be expected. From these beginnings, the use of the infinitive with to in place of the simple infinitive, helped by the phonetic decay and loss of the inflexions and the need of some mark to distinguish the infinitive from other parts of the verb and from the cognate n., increased rapidly during the late OE. and early ME. period, with the result that in mod.Eng. the infinitive with to is the ordinary form, the simple infinitive surviving only in particular connexions, where it is very intimately connected with the preceding verb (see below). To a certain extent, therefore, i.e. when the infinitive is the subject or direct object, to has lost all its meaning, and become a mere ‘sign’ or prefix of the infinitive. But after an intrans. vb., or the passive voice, to is still the preposition. In appearance, there is no difference between the infinitive in ‘he proceeds to speak’ and ‘he chooses to speak’; but in the latter to speak is the equivalent of speaking or speech, and in the former of to speaking or to speech. In form, to speak, is the descendant of OE. tó specanne; in sense, it is partly the representative of this and largely of OE. specan.
(The simple infinitive, without to, remains: 1. after the auxiliaries of tense, mood, periphrasis, shall, will; may, can; do; and the quasi-auxiliaries, must, (and sometimes) need, dare: 2. after some vbs. of causing, etc.; make, bid, let, have, in sense 15a; 3. after some vbs. of perception, see, hear, feel, and some tenses of know, observe, notice, perceive, etc., in sense 15b; 4. after had liefer, rather, better, sooner, as lief, as soon, as good, as well, etc.: see HAVE v. 22, RATHER adv. 9d, and the other words.)
The infinitive with to may be dependent on an adj., a n., or a vb., or it may stand independently. To an adj. it stands in adverbial relation: ready to fight = ready for fighting; to a n. it stands in adjectival or sometimes adverbial relation: a day to remember = a memorable day; to a vb. it may stand in an adverbial or substantival relation: to proceed to work = to proceed to working; to like to work = to like working.

Now they’ll be posting!  In fact, they’ll be falling over themselves in their rush to pick holes.  Not to worry - I’ve had so many holes picked in me already that I’m like a piece of wire netting.  Which, of course, may also be barbed.

How could I resist such an elegant gauntlet?

Actually I don’t have much concrete. Many sites differ and I haven’t found any that I’d care to firmly hang my hat on. Some say it’s a ‘fossilized preposition’, whatever that is when it’s at home. Some say it’s a particle (which seems to be a catchall term for any part of speech that doesn’t fit elsewhere.)

Yer pays yer money etc etc

I think most would go with particle. But as aldi says, there are different opinions and there is no “right” answer.

Yeah, preposition doesn’t seem quite right. It’s not doing what a preposition is meant to do.

Might have to wear “particle”, which is as much as to say (if I may paraphrase Prince Regent George) “Well, it doesn’t really mean anything, does it?”

EDIT: removed superfluous Well

There are more google hits for “infinitival particle” than “prepositional particle” or “particular preposition”. 

So, if I can try to put it in my own words (the words of a lay person), in the beginning English didn’t have this particular preposition — to — to mark the infinitive. It had instead, besides a “plain” (invariable?) infinitive, a variable (?) infinitive that was used to form (mostly or always?) nouns. (Is this particular noun what is then called a “déverbal” in French? (“verbal noun” in English?) e.g. “le boire et le manger”, or “portage” from the verb “porter”.) And to this noun the preposition to was later added to form the “modern” inflected form of the infinitive?

That reminds me of an old skipping rhyme: Keep the kettle boiling and never miss a loop…

"That reminds me of an old skipping rhyme: Keep the kettle boiling and never miss a loop… “

How so?

Oh, dear. 
might help… to keep the discussion going/kettle boiling/not missing an opportunity to do so ...
Please don’t ask for further explanation.

Okay, I won’t.

Meanwhile, thanks, Eliza, for that history of the English infinitive.

Yes, most informative.

You’re welcome.

Interestingly, Swedish has a parallel construction for the infinitive, with “att” in place of English “to”. My knowledge of historical Swedish doesn’t stretch to the grammatical origins.