I don't have a precise idea of what my dream incremental game is. There are usually minor changes to it from one day to the next. However, there are a few traits that pop up most of the time. I'll list them down, in no particular order.
Long-term and 'open-world' - I like the feeling of discovering mechanics and how to use them in incremental games. I like it when an incremental game allows me some flexibility in my play style. I generally want to feel like my save file is 'mine', that it represents a journey that is uniquely mine.
Subtle storytelling and thematic cohesion - This is good in general. However, the thing I'm particularly looking for is a sort of story-focused representation of progress - that as I progress I'm exploring more of a world or growing something.
'Easy to learn, hard to master' - My dream incremental game tends to have a decently sized set of mechanics introduced at the start, and accumulates new mechanics very gradually. However, there is lots of room for optimization and discovery with just the core mechanics. Each newly introduced mechanic should act as a natural extension to existing mechanics, broadening the range of possibilities and altering your optimization process (so, paradigm shifts).
Small numbers - The game shouldn't need to use big number libraries. If no numbers ever have any chance to go beyond a quadrillion (which I consider dangerously close to Number.MAX_SAFE_INTEGER in JavaScript), even better.
Distinct art style - Even just simple graphics will do, that makes it stand out from most incremental games already simply by virtue of existing. Just don't let it
Either really idle or really active - It should either be really active in the sense that no progress can be made without direct player interaction, or really idle in the sense that the game can still progress at a significant rate can even after a month or so of idle time (Still allow some player interaction, of course. See Gooboo for an example of what I'm trying to convey.)
Thank you for the lengthy reply, it was a fun read!
Slowly unfolding mechanics are nice, it is not enough to be overwhelming but I still feel like I'm making progress and learning new things. I also like discovering things and how to use them, although we might disagree on just how many mechanics to have at the very start.
I definitely get and share the love for build diversity, multiple valid strategies and ways to play…
Stories in incremental games are super cool, whether they are an explicitly narrative-heavy game or it's something more like Shark Game—not exactly super narrative-driven but those cute pixel arts of a shark starting to make a whole civilization down there complete with flavor text… there is definitely a story there. I also feel like Kittens Game kind of did it, I was imagining the civilization the kittens made from time to time, thinking about their progress…
Shark Game stands out because of the adorable art. I'll be honest, I'm a very visual person and good-looking games catch my attention very quickly. I definitely don't need the kind of graphics I need a NASA computer to render, it's more about the style, really… if you are a pure menu UI it can still be attractive to me!
Aside from "really active", I feel I at least somewhat agree with you on most of these points.