What does 'ages of ages' mean, specifically? I run into this phrase a lot in the Orthodox church, particularly as part of the longer phrase 'and unto ages of ages'. I understand that, generally, this means 'for a long time'. However, what does it mean specifically? Where did the phrase come from and what is its history? Thanks!
The phrase comes from the Bible. It’s a fairly wooden translation of the Greek idiom εἰς τοὺς αἰῶνας τῶν αἰώνων. Many English translations instead use “forever and ever” or something along those lines, which is more in line with English idiom. 

As for precisely where it originated, I haven’t been able to find any uses in classical Greek, though perhaps someone with better searching abilities can help with that.  Liddell-Scott-Jones Greek-English Lexicon (a lexicon including classical Greek) discusses several related idiomatic phrases using a double αἰών with various prepositions. Interestingly, all of the examples given there are within the LXX. 1 

Εἰς τοὺς αἰῶνας τῶν αἰώνων itself appears just once in the Greek translation of Hebrew scriptures,3  in Psalm 83[84]:5[4]2 where it translates really nothing (עוֹד ≈ still or yet), so the idiom appears to be native to Greek. It shows up 18 times in the New Testament (NA-28, including bracketed):


Galatians 1:5, Philippians 4:20, 1 Timothy 1:17, 2 Timothy 4:18
Hebrews 13:21
1 Peter 4:11
Revelation 1:6, 1:18, 4:9, 4:10, 5:13, 7:12, 10:6, 11:15, 15:7, 19:3, 20:10, 22:5


All of the examples from the epistles are contained within a doxology, a fitting term because all of them ascribe δόξα (doxa ≈ “glory") to one or more persons of the Godhead, with the basic structure: ᾧ ἡ δόξα εἰς τοὺς αἰῶνας τῶν αἰώνων (“to whom be the glory forever and ever”).

The phrase appears to have been pulled from these Biblical doxologies into Orthodox Liturgies. You’ll also run across the Latin version of the same: in saecula saeculorum.  



 
1. Koine Greek witnessed increased laxity in the meaning of prepositions and tended to add them in where classical Greek found the naked cases sufficient, so it’s not especially surprising that the development would go in this direction.
    


2. Numbering is according to LXX with English in brackets. 
  


3. And once additionally in the Greek 4 Macc 18:4.