I am often making a basic leavened bread for some basic recipes, like naan, burek, focaccia, pizza, etc. However, I find that dry active yeast is prohibitively expensive, considering it is a self-replicating living organism. Is there a way to effectively create and store yeast for a reasonable period of time (~3 months) from a mother spore?
In order to store yeast for that period of time it has to be make inactive, and the process to do that yourself is challenging, time consuming, and you'd need specialist equipment. It would be prohibitive in both time and cost, far above just buying yeast.
You can of course make a starter by mixing equal amounts of flour and water plus some yeast, this will keep the yeast alive by feeding it, and you can keep it alive indefinitely by feeding it once per week. You add some of the starter to your bread and it supplies active yeast.
You can "hibernate" a yeast starter in various ways.
Firstly, if you leave a starter in the fridge, and neglect to feed it, it can be revived after months or even years, by kneading it into some fresh dough, and feeding it as normal for a few days.
Secondly, you can freeze a starter. After thawing it, again knead into fresh dough, and feed as normal, and once again you'll have a healthy starter after a few days.
Thirdly, you can preserve a starter by drying it. Spread some starter very thinly on parchment paper, and allow it to dry completely. Crumble this into fresh dough to create a new starter containing the old yeast.
If you bake regularly, though, you don't need to do any of this. Just keep a starter continuously fed. Never use all of it at once, but stir in some fresh water and flour to replace what you've used.
If you're lucky, or careful, you can preserve the strain of yeast you start with. Normally, however, even if you start with commercial yeast, a wild strain will soon become dominant, and what you have there is a sourdough. Many people prefer sourdough, but you do have to get used to its slower action and its more acidic flavour.
Preventing other strains from colonising, is one of the complexities that contributes to the cost of producing commercial dry active yeast. However, as others have noted, bought in tubs rather than sachets, it should be very cheap compared to the other ingredients in your bread.
You can also try to capture "wild" yeast. Place a glass with water and flour just covered with some fine mesh (so no insects or large particles fall into it) and leave it overnight exposed out. Some yeast you catch would be terrible for making bread but some others would yield a great sourdough. Once you find you you like you can keep it alive forever in the manner answer 1 describes.
I would also recommend going for the "starer idea", using water, flour and a little yeast. I did this last year for severeal months and it worked really well for me (used it to bake bread most of the time).
However, I just wanted to add, that I know this kind of starter-usage as sourdough, rather than yeast.