Make sourdough starter without discarding

You need the flour to feed the bacteria. The amounts given in the starter-process mean that the colony of bacteria will be happily fed and able to grow until the discarding and filling up.

Your starting-amount will simply not keep the colony happy and fed for the required time. Also, with that small a mass, chances are the starter would dry out until the next "feeding".

I won't say your idea doesn't work, but there are a few points to consider:

A "ripe" sourdough contains a balanced culture of yeast and bacteria (lactobacillus) that can keep fungi and other unwelcome bacteria under control.

Let me use the political landscape of today's US as an example:
You have two big parties in slightly varying majorities - like in a sourdough that may be a bit more yeasty or a bit more acidic depending on the circumstances. But together that system keeps extreme political groups that may topple over the whole system from rising to power - like in your sourdough.

Now if you take a very small sample and especially if your two parties (yeast and bacteria) aren't fully established yet, even a comparatively small number of extremists (mold, other bacteria) can endanger the whole system.

So if you want to start a sourdough from a tiny amount of flour, you need to work very carefully and cleanly to avoid contamination and still might have too few yeast spores etc. in your initial amount of flour to get to that stable system going. Technically, it is possible, though.

Larger amounts of the initial flour/water mix increase the probability of successfully getting a sourdough going - most sources I know suggest around 50g / 2oz of flour as minimal initial amount for home bakers. While I respect your frugality in not wanting to discard anything, a few spoonfuls of flour are pretty cheap and if you have to start over a few times, you'll gain nothing.

While I have successfully begun a starter with only 10 grams of flour at the outset, I think I agree with Stephie that doing a little more is more efficient. I used to do very wasteful starters until I happened upon this site back in 2002 or something. (Amazingly, it's still there.) You can read the details there, but basically the guy who did it designed his initial starter regimen based on the amount of flour he'd eventually use to bake bread (i.e., with no waste). (It also has a way to get to a lower pH earlier which helps the starter along while inhibiting bad stuff; this was years before the internet discovered the "pineapple juice solution.")

But I think this depends a LOT more on your feeding regimen, especially in the early days, as well as the flour. I've written a lot about the variables in my other answers here on sourdough related questions, so I'm not going to repeat it all here.

Briefly, though, YES, it is possible. It can be just as reliable as using more flour and a lot less wasteful. Even if you don't do a true exponential growth model for your starter, you can also be significantly less wasteful by keeping the starter relatively small until it's established. There's no reason to be feeding and throwing away a cup or more of flour every time for every feeding, particularly if the starter isn't even growing yet.

Also, the same applies to maintaining your starter. If you want to do detailed calculations, I've explained them here. When I first started with sourdough, I used to produce way too much of it and throw it away. Now I generally save just a tablespoon or two, feed it a little, and put it away for the next batch and build up from there when necessary.

In the beginning of a starter's life, the point is to develop yeast and friendly bacteria, NOT develop lots of starter. If you don't discard, you end up with way too much, and it is not wasteful because tossing about half keeps everything at a manageable amount. You don't want too little either because as was mentioned, there needs to be enough there to keep it fresh and good. The pioneers coming access America in covered wagons had no refrigerators yet their starters did good because they were populated with wenough yeast and bacteria to stay that way. Everyone suggests about a cup of starter, a cup of fur and either 1/2 cup or 2/3 cup water, and this has worked well for me. I know my answer may have seemed a bit contradictive, but too little or too much should be avoided.

Starter becomes acidic over time. You need to discard and replenish to keep acid under control. Too much acid inhibits yeast growth...bad for your loaves!

It's not clear from the question whether the OP asks about the starter culture or the mother starter. The answers so far seem to pertain to the starter culture; being small and done only once, the quantity of discard from this process is not very meaningful. Discard from repeated uses of the mother starter - the starter used to raise and flavor dough for baking - is done frequently with larger amounts, so it is the more meaningful question.

I bake bread at home for one person, so the amount of mother starter I use each time does not vary greatly. By "bread" I mean loaves, muffins, pizza, rolls, etc. Knowing what I'm going to bake next, I replace the amount of mother starter just removed to make a dough with a slightly larger quantity of flour-plus-water than I will need for my next recipe, to account for starter "lost" to cleanup, etc. This process ensures that I am using about 50g - 100g of *mother starter" each time, leading to repeatable re-growth of the mother starter.

The SCOBY in a sourdough starter is much more forgiving that some would have you think. Remember: Nature is a mother - all of her creatures want to live and are likely to find a way to establish themselves in your starter as long as you follow basic rules of cleanliness and don't expose them to chemical challengers.