How many are there of God's angels?

None of the major branches do. All they say is that it is a large number, and most say that there are more than all humans who have ever existed:

A Protestant view: (Drawing from Scripture only)

While the Scriptures give no definite figures, we are told that the number of angels is very great (Daniel 7:10; Matthew 26:53; Hebrews 12:22).

It appears that all angels were created at one time. No new angels are being added to the number. Angels are not subject to death or any form of extinction; therefore they do not decrease in number.

It seems reasonable to conclude that there are at least as many spirit beings in existence as there will have been human beings in all their history on earth.

Author: Dr. Paul Eymann.

A Catholic view:

The number of the angels is frequently stated as prodigious (Daniel 7:10; Apocalypse 5:11; Psalm 67:18; Matthew 26:53). From the use of the word host (sabaoth) as a synonym for the heavenly army it is hard to resist the impression that the term "Lord of Hosts" refers to God's Supreme command of the angelic multitude (cf. Deuteronomy 33:2; 32:43; Septuagint). The Fathers see a reference to the relative numbers of men and angels in the parable of the hundred sheep (Luke 15:1-3), though this may seem fanciful. The Scholastics, again, following the treatise "De Coelesti Hierarchia" of St. Denis, regard the preponderance of numbers as a necessary perfection of the angelic host (cf. St. Thomas, Summa Theologica I:1:3).

An Orthodox view:

St. Cyril of Jerusalem writes: "Imagine how great in number is the Roman people, imagine how great in number are the other barbarian peoples that now exist, and how many must have died even! In a century, imagine how many have been buried in a thousand years, imagine all mankind, from Adam to the present day. Great is their multitude, but it is small in comparison with the angels, whose numbers are greater. They are the ninety-nine sheep, whereas the human race is the one lost sheep. By the greatness of a place one can judge the numbers of those who dwell in it. The earth we inhabit is a mere dot in the heavens, thus the heaven that surrounds it must have a much greater number of inhabitants. As is has greater space, the heavens of heavens hold their innumerable number. If it is written that 'a thousand thousands ministered unto Him, and ten thousands of myriads attended upon Him' this is only because the prophet could express no greater number." When the numbers of the angels are so great, it is natural to assume that in their world, as in the material: world, there are various degrees of perfections and therefore various ranks or a hierarchy of the heavenly powers. Thus Holy Scripture calls some angels and others archangels (I Thess. 4:16, Jude v. 9).

There are a great many teachings of the number of types or ranks of angels, but no serious writings about the population count.

Of course, there may be smaller groups within Christianity that try to pin a number on it, but most of Christianity would discount any such attempt the same way we'd discount all of the various attempts to set a date of the second coming. It wouldn't be accepted by Christianity as a whole.

The only exception I can think of - the only group within Christianity that puts an actual number to it is the group of "Christian Atheists" that discount all supernatural altogether. Those in that group probably put the number at zero. But that is a small group that does not represent any of the major denominations or teachings (nor do they believe in them except, perhaps, the moral ones they agree with).