Four is a good limit. Most even semi-sentient things should understand something sliced twice is "quartered" into four, for example.
If you read any paragraph of well-written text, you are very unlikely to find one which has a specific number larger than four, other than in threads like this where numbers are being explicitly described.
In normal speech, we use numbers very rarely. We give far vaguer terms, instead. This is unlikely to be an issue for you.
Consider comparing opposing forces:
- There's just a family of them, we can take them.
- There are rather a lot of them, this might be risky.
- There are more of them than us, but we might hold if we dig in.
- There are a lot more of them than us, they'll overrun us.
- The plains are crawling with them, we must run now!
Many of our collective terms have a built-in size limits which are approximate, non-numeric, but very well-understood by everyone who uses them. People are grouped as self, family, clan, tribe, nation, race. Property as desk, room, household, street/block, village, town, county, country, continent, planet, solar system, galaxy, universe. A pod of dolphins is up to a few dozen individuals; above that it's a group, then a supergroup for when it's up in the hundreds. Same with distance: a thumb, foot, pace, walk, long walk. Money: a farthing, ha'penny, penny, tuppence, shilling, quarter, pound...
Time goes in moments, heartbeats, breaths, morning/dawn/afternoon/evening/dusk/night (do we have a name for these day-parts?), *day/yesterday/tomorrow, moon, season, year, reign, epoch. You don't need dates like "1986", you just need "towards the end of the glam-rock era", "a few years before millennials started being born".
Containers and ingredient quantities are a pinch, thimble, teaspoon, tablespoon, cup, fist-sized, head-sized, crate,... you never need to measure an ingredient as "a head-sized lump plus two pinches", you're good so long as you're within 20-50%. So as long as your measures are within a couple of sizes of each other, you can make it work just by using them. Which is basically how cooking in America works, in fact, with tsp, tbsp, floz, cup, pint, quart, gallon... replace those with "apple sized" and such, and you've got something which works for any society.
Even generic terms, like a couple, few, several, lots, plethora, lots'n'lots, a shitload, countless, overwhelming ...have comparable (if overlapping) sizes.
Before you knew what a score or a gross or a dozen or a hogshead was, you got some sense of the scale of the thing from context. "I only asked for a few score of them, but they gave me a whole gross! How can I ever carry that many eggs home?"
So, give them a plethora of vague measurements, rather than numbers. If they have reason for more specific numbers in the dialog you are writing for them, then that means they have a need to measure stuff more carefully than their approximate grouping nouns permit them to. That in turn means they are at a technological stage where that matters, and your rationale for having them not use numbers no longer holds up, and probably needs reexamining in the context of your world.
Edit: Since this became the accepted answer, I thought it best to add a little hard data in here. So, to give some measure of how rare exact numbers are in common writing, I gathered from the Gutenberg project a few books and checked them to see how many numbers they contained, compared to how many words.
So, appropriate to the discussion, here are some numbers!
- Brahm Stoker's Dracula: 359 numbers in 148,036 words.
- Peter Pan: 88 numbers in 42,928 words.
- War and Peace: 1,169 numbers in 512,229 words.
- Pride and Prejudice: 147 in 110,865 words.
From this, we can see that numbers appear about one time in 500 to 1000 words.
Reading through, I found that many seemed to be used to describe time (people's age, time of day, number of minutes' wait, etc); distance (twenty feet away"); currency ("ten pounds"); and as identifiers in addresses ("at number 27").
Of the remainder, many were used vaguely "five or six", or as superlatives when it was clear that the speaker was not actually confident in the number ("must have been twenty of them!"). Others were used specifically to make the reader's eyes glaze over: one third of the numbers in Peter Pan appear in a single passage with the father trying to do math out loud and getting it wrong.
I was very hard pressed to find a case where a precise number was specified as something other than an identifier, and where the precision would actually matter, to the speaker, listener, or reader.
[For anyone choosing to reproduce or improve on this rough test: I used the plain text versions of the books, stripped off everything above the beginning of the first chapter (contents page, etc), and everything from the last line of the original book (Gutenberg license etc). I then globally regex-replaced the term
chapter \d+ with 'Chapter N'. To find the numbers, I searched for the regex
(?!\b[0-4]\b)\d+|\b(five|fifth|six|seven|eight|nine|ninth|tens?\b|tenth|eleven|twelve|twelfth|thirteen|foureenth|fifteen|twenty|thirty|fourty|fifty) - to find the words, I searched for
+ (one or more spaces)]