A long way - but you'll probably learn to count eventually by accident
You're going to have to try really hard to not count!
Counting is a very specific action in mathematics - integer incrementing. 1, 2, 3, 4, etc. From incrementing integers we can derive most of mathematics, however that need not be the source of it. I'm guessing perhaps alien biology here (no fingers to count on?)
If we start with Boolean logic: True / False. And from that equality and inequality (True != False) == True. And then Boolean operations (true and true == true, true and false == false etc). We could step into probabilities, giving us the constants Always and Never, and giving us a sense of ordering before giving us a way to represent numbers (Eg probability of rain this year > probability of earthquake this year).
From the definition of equality with regards to probabilities we could define "Fair" as equal probabilities of both outcomes happening. We could then subdivide the probability space from Never...Always into equal chunks, so we basically have recursive power of -2 constants, giving us fractions with power of 2 denominators, eg: 0, 0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.875, 1.0). You've basically discovered binary fractions.
From binary fractions and known logical operations you can discover addition, subtraction, and multiplication. The discovery of addition doesn't mean you've discovered counting yet, as we haven't discovered integer representation yet - 0.25 is just "halfway between Never and Fair.", lets say they represent it as FN for short (Start at Fair, then halfway to N). 0.5 is F. 0 is N. 1 is A. FAA is 7/8 (Start at fair, half to A, then half that again towards A). FAAN is 13/16. FAAAAAAAAAAAAAA is 65355/65356 (0.99995409755)
From multiplication you can discover division. You can discover square roots from self multiplication. From addition and subtraction you can discover numbers below "Never" and numbers above "Always". From division you can discover square roots, and from square roots you can propose "imaginary" numbers.
You can start applying this probability to the real world via geometry. Draw a circle that touches all 4 sides of a square, what's the probability of a random selected point in the square being also inside the circle? From that and other circle probabilities you can get an approximation of pi.
You could even get to algebra - by identifying an unknown value and recording it as such you could do all sorts of things.
I think you could do a lot of maths from this approach - basically all of it. You only need to "count" when your applying probabilities to collections of real world things. Where this kept entirely theoretically I see no reason why you couldn't get calculus, trigonometry, etc before discovering counting.
You can measure distance by making a ruler the length of a known thing. A is 1 ruler long. N is 0 rulers. F is half a ruler. If the ruler is 1m long, I'd be A + FANAN rulers tall. This is getting a bit of a grey area as determining that my farm is 431 rulers long is easiest done by incrementing an integer as I move the ruler. This is one of many ways in which you'll inadvertently discover counting.
Currency could work - especially if like bitcoin your just dealing it fractions of a big value, Eg 1.0 = "All the money in the kingdom", for example. But buying and selling goods is like ground zero for counting.
I think government will discover counting while trying to distribute resources, as in order to calculate the fraction of resources a group gets, one must count the groups size. Or some other Census. Trade and Taxation could theoretically work under this number system but opportunities to count would be everywhere.