You're right that there was less common need of very large numbers (at least so far as the central latin middle ages are concerned), but they definitely had a conception of such numbers and did in fact use them on occasion. The example that springs to mind is that ancient and medieval authors had estimated what they thought to be the diameter of the universe. For example, here is Roger Bacon's estimate from the mid thirteenth century (Opus majus, ed. Bridges, 1.227):

Et si duxerimus diametrum in rotunditatem, erit superficies totius coeli stellati, 53,700,149,508,265,000, videlicet, quinquagesies ter millesies millesies millesies millesies mille millaria, septingentesies millesies millesies millesies mille milliaria, centies quadragesies novies millesies millesies mille milliaria, quingentesies octies millesies mille milliaria, ducenties sexagesies quinque millia milliaria.

And if we will consider the dimeter on the circle, the surface of the entire stellar heaven will be 53,700,149,508,265,000, that is, fifty three thousand thousand thousand thousand thousand miles, seven hundred thousand thousand thousand thousand miles, one hundred forty nine thousand thousand thousand miles, five hundred and eight thousand thousand miles, two hundred sixty five thousand miles.

There are two particular points of note here, first this is after the introduction of Arabic numerals, which Roger naturally uses for large number, and second how unwieldy it is to express this in latin as he doesn't seem to have a term for a number higher than 1000, nor another convention for straightforwardly expressing such a long number.

Astrology is an excellent place in general to find large numbers throughout history. Time and space on a universal scale (depending upon the universe concept) calls for them in this case. I found something from the 3rd c. covered by Hand (2000) 'On Astrology, Prophecy, and Religion' Revision, 23, pp.38:

In particular, Berossus appears to have taught that the time from creation until a universal catastrophe would be 2,160,000 years. This consisted of five groups of 120 Babylonian Sars of 3,600 years each, or 432,000 years per group (5 x 3600 x 120 = 2,160,000). This is an interesting figure because it is exactly one half of the Hindu Mahayuga of 4,320,000.

The Mesoamerican long counts and other such records appear to reach very high numbers, though the words for them are later constructs. J. Thompson did a lot of work on Mayan calendar systems, for reference.

The largest number that the ancient Greeks had a word for was 10,000, which they called myrias. This also doubled as a non-specific indication of a large number of something. In fact, their word for 10,000 is still used in modern English to signify "a lot" - a myriad.

There were some things that required them to describe numbers significantly larger than 10,000. The Greeks lacked a measure of distance greater than a stade (200m), and often expressed distances in clumsier units such as the cubit (50cm) or the orguia (about 180cm), so that significant distances (like the span of a sea or the circumference of the Earth) had to be expressed in ridiculously large numbers of such units. In addition, wealth (number of coins), produce (amount of bags/bushels), or Persian armies could be far larger than 10,000.

The way to express such numbers was the way that is still used in Japan and China - simply use multiples of 10,000. Herodotos (7.28.2) expresses the wealth of a particular Lydian as "in gold, four hundred times ten thousand Daric staters, short seven thousand" (= 3,993,000). He claims that the full invasion army of Xerxes, including the numbers of the army, fleet, supply train, engineers, servants, eunuchs and camp followers, was

Five hundred and twenty-eight times ten thousand and three times a thousand and two times a hundred and two times ten.

-- Hdt. 7.186.2 (= 5,283,220)