Vjylku

There are a few instances in Latin where words are spelled with two vowels next to each other, in hiatus: filii "sons", metuunt "they fear".

Now, the last words of the Emperor Julian II are normally quoted as vicisti, Galilaee. The second word here is the vocative of Galilaeus, "man from Galilaea".

However, the ending -ee looks distinctly wrong to my eye. Is this sequence ee something found in Classical Latin? Or does it only appear in Later Latin? (Notably, common words like meus and deus don't take vocatives in -e.)

First, Galilaee sounds right.
See this question about the vocative of Gnaeus for details.

There are situations where one finds -ee- in Latin without the first e belonging to ae.
What I found is not word-final, but I assume that is not important for your question.
There are forms of deesse and deerrare, and if the diphthong ae is included, also forms of praeesse.
There are lots of examples; you can see the dictionary entries for these words or make a corpus search.

In addition, there are first conjugation verbs ending in -eare, and in conjunctive it leads to forms like crees and recreetur (Cicero, In Q. Caecilium and Pro Quinctio).

It seems that one can legitimately form the Latin wish beer ("may I be happy").
I found no attestations of this, but beet does exist.

The Vulgata is full with proper nouns having double -ee, specially as endings (e.g. Bersabee, Phacee, Osee). I imagine you are not particularly interested in these. Below are all the other words I could find:

• 
  deest, deerunt, deessent, deerit, deerant, etc. E.g.

Nm 21:5 locutusque contra Deum et Moysen, ait : Cur eduxisti nos de Ægypto, ut moreremur in solitudine ? deest panis, non sunt aquæ : anima nostra jam nauseat super cibo isto levissimo.

Dt 15:11 Non deerunt pauperes in terra habitationis tuæ : idcirco ego præcipio tibi, ut aperias manum fratri tuo egeno et pauperi, qui tecum versatur in terra.

• 
  eleemosynæ and related. E.g.

Tob 2:22 Ad hæc uxor ejus irata respondit : Manifeste vana facta est spes tua, et eleemosynæ tuæ modo apparuerunt. (also in several other verses)

• 
  reexspecta. E.g.

Is 28:10 Quia manda, remanda ; manda, remanda ; exspecta, reexspecta ; exspecta, reexspecta ; modicum ibi, modicum ibi.

• 
  procreentur. E.g.

Tob 6:21 Tertia autem nocte, benedictionem consequeris, ut filii ex vobis procreentur incolumes.

• 
  illaqueentur. E.g.

Is 28:13 Et erit eis verbum Domini : Manda, remanda ; manda, remanda ; exspecta, reexspecta ; exspecta, reexspecta ; modicum ibi, modicum ibi ; ut vadant, et cadant retrorsum, et conterantur, et illaqueentur, et capiantur.

• 
  spontanee. E.g.

Os 14:5 Sanabo contritiones eorum ; diligam eos spontanee : quia aversus est furor meus ab eis.

Some of the above have other declensions that are not in the Vulgata, but also have double ee. For instance, subjunctive passive declensions of procreo.

Regarding the period, some seem to be Classical (e.g. procreo), with others being Late Latin (e.g. eleemosynæ).

Rarely ee can be used as geminatio vocalium, i.e. to denote that the e is pronounced long. This was mainly used in Oscan and sometimes borrowed to Latin.

For example leege in this inscription:
Vediovei patrei / genteiles Iuliei // Vedi[ov]ei aara // Leege Albana dicata

(aara also denotes long a).

M. Loporcaro in Vowel Length from Latin to Romance notes that Accius (c. 170-90 BC) recommended to write long vowels by double letters. F. Rovai (in the context of grave inscriptions) mentiones that this practice was chiefly restricted to either official epigraphy or poetic, archaising epitaphs.