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In Greek, past tenses are formed with "augmentation," e.g. present -> imperfect:

  • λῡ́ω > ἔλῡον
  • εὑρῐ́σκω > ηὕρῐσκον

Since we know that certain Latin verbs preserve perfect reduplication, I wonder: do any Latin verbs also preserve a temporal augment? One potential example immediately comes to mind: ago > egi.

If so, is there any rule we can point to by which some Latin perfect stems are reduplicated, others are augmented, and the rest are formed according to the normal rules?

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First, it's not at all clear that Proto-Indo-European used the augment (temporal or syllabic): it's found in Greek, Indo-Iranian, Armenian, and Phrygian, but not in the other languages, so most likely it was not an obligatory marker of the imperfect and aorist in PIE, if it was used at all.

As for Latin, as far I know, there's no evidence it ever used an augment. (I should maybe say "almost no evidence" because I have a vague memory that there is an archaic inscription in some Italic language which has been argued to contain an augment, but I can't remember the details at the moment, and that interpretation is contested.)

The explanation of ēgī, and other perfects with long ē, is a debated subject; no one to my knowledge has proposed that ēgī contains an augment, partly because this would divorce it from other such verbs where the ē is not initial (e.g. lēgī, sur-rēgī), and it would be preferable to have a unified account of the entire class. (In any case, there is no Attic-Ionic-style change of ā to ē in Latin, so a temporal augment would not yield ēgī.) The theory given in Weiss (2009:413) is that these reflect PIE imperfects of the so-called "Narten class" verbs: these are verbs that show a different ablaut pattern from normal PIE verbs, including a lengthened grade where other verbs have a full grade; in the imperfect, this means such verbs had a long ē.

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  • It’s probably worth mentioning that in the case of agō, it is the a that is secondary in PIE, not the e. The underlying root is *h₂eĝ-, and PIE /e/ was phonetically lowered to something like [ɐ ~ a] after /h₂/ (phonetically probably [x] or similar). In the Narten imperfect (let's assume that's what it is for simplicity), however, the vowel was /eː/, which was not lowered after /h₂/. So phonetically it would have been something like [xɐɡʲ-] in the present, [xeːɡʲ-] in the preterite.
    – Janus Bahs Jacquet
    Commented Mar 4, 2017 at 10:33
  • Also, if it were an actual augment, I believe it would indeed yield ēgī without the need for an Attic-Ionic-style change of ā to ē, since *e+a in Latin regularly yields ē, not ā. If the augment was indeed a (Late) PIE thing, it was likely—at least going by available evidence—considered separate enough that it did not lower to [a] in the vicinity of /h₂/.
    – Janus Bahs Jacquet
    Commented Mar 4, 2017 at 10:38
  • @JanusBahsJacquet, that's a valid point about e+a contraction; I was addressing the specific question about a temporal augment, i.e. vowel lengthening rather than e- prefixing. (Of course, Latin has no temporal augment, so there's not much point in speculating about what it would have looked like.)
    – TKR
    Commented Mar 4, 2017 at 23:07
  • @JanusBahsJacquet, by the way, what evidence do we have for whether the augment was colored by laryngeals?
    – TKR
    Commented Mar 4, 2017 at 23:30
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    Precious little, if any at all. I thought I remembered that there was some evidence, but I can’t find what it was I was thinking about now. Greek isn’t worth much here, of course, since the contractions are later (so even where PIE would have had an actual consonant between the augment and the root, e.g., *e-seĝʰ-, Greek contracts into εἴχ-). There are some marginal cases like Skt. यभति yábhati, impf. अयभत् áyabhat, where the root is *h₃i̯ebʰ-. If regular, you’d expect PIE *e-h₃i̯ebʰ- > *o-h₃i̯ebʰ- > IIr./Skt. *ā-yabʰ-. But that could so easily be analogically levelled.
    – Janus Bahs Jacquet
    Commented Mar 6, 2017 at 18:20
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This is essentially a partial answer, but you're asking two questions so maybe someone else can provide the other half. I can only reply to your first question:

It seems to be exceedingly rare, if it occurs at all. The only other possible occurrence I could think of, and I wouldn't have thought about it if you hadn't mentioned ăgo > ēgi, is ĕmere > ēmi. But this train of thought is compromised by such cases as făcere > fēci or vĕnire > vēni, where a similar phenomenon occurs in Inlaut. You're basically stuck looking for irregular root forms (without prefixes !) of verbs that start with a vowel, and I am inclined to conclude that your initial premise of ăgo > ēgi as an augmented root is essentially incorrect.

One reason why I can see e- augmentation not staying a language feature in Latin, if it was inherited from PIE, is the verbal prefix ex-, which becomes e- before most consonants. This would introduce huge ambiguity between a lexical shift in meaning, and a grammatical one. I don't know any other languages from antiquity, so I'm not sure whether there is a similar prefix in other languages that might invalidate this point.

Since I don't believe there is augmentation in Latin, part of your last question is moot in this answer. But I can't seriously reply to the rest of it because I don't remember any rules why some words would get reduplication and why some wouldn't.

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