What purpose does the Fortune Teller serve?
So I'm assuming the Fortune Tellers predictions about chance impact some RNG in the game, but I'm really not sure what RNG it impacts, since there is quite a few things I can think of (chance for rare crops, chance for minerals to appear in the mines, chance for fish to bite, et cetera).
So what do the Fortune Tellers predictions impact?
Geodes - there were threads all over claiming their contents was impacted by luck. However Tallian linked this thread on reddit in the comments that suggested otherwise. The OP could be better worded, so I'll restate. What you get out of Geode's appears to be deterministic. I did a very small test last night and opened a couple of geodes, force closed the game, and then opened them again. I got the same set of things in the same order both times. So it appears Geode content is set in some manner. It also might be true that it's deterministic by type, as per this comment in the thread (essentially, that the type of geode + the "nth Geode opened" is what determines the drop). Unfortunately I only had basic Geodes on hand for my test. There have been arguments that what a Geode contains is set when the Geode drops. However nobody has found evidence in the save files, which contain lots of information about the things your character possess. So I would say Geode contents are not affected, but Geode drop rate probably is.
Drops in the Journey of the Prairie King arcade game in the Saloon. I haven't played the other game but if it has drops then it affects those as well.
What you find in the mines. A good fortune means you are more likely to find a stair under a rock. I believe it impacts the spawn rate of rocks with ore/minerals in them as well.
Fishing.
Obviously a bad fortune works against you while a good fortune works for you.
Crops @JohnDubya says he has read that fortune impacts crops. I can't offer much in the way of evidence on this one, but here goes. I've put something like 14+ seasons of crop growing into this game (less a couple of winters without the greenhouse) and as I tend to harvest as soon as possible and use quality fertilizers I can't speak much to whether luck might be impacting harvesting. In my current game's green house, I only use speed growth fertilizers (I'm increasing my base for star, ancient, and rare seeds). I pulled two ancient fruit the other day and got a silver and a gold despite it being a worst-luck day. I almost always get some quality out of the crops there and I know I have harvested on a series of bad luck days over the past in-game month (as I'm just building up a seed base I don't have any steady crop rotations; most times I harvest in batches of 2-6 plants). In my very humble opinion, waiting for a "good luck" day to harvest is a bad idea even if it impacts the quality of what you harvest unless one is well into the game and has a terrific economic base. Plant with Quality Fertilizer instead and there will be plenty of gold starred crops.
I find it much easier to find stair down in the mine on my fortunate day, and the opposite when not. The monster spawning is affected too. I can descend some 5 floors in a few hours when my luck is high, while it can take entire day and got killed on my non-lucky day.
I think the fishing is also affected, but I haven't been keeping track much in this case.