Apparently not many anthropologists or people interested in history on Lemmy.

There’s a few options, and it depends on what you mean by “gods.” The overall category you’re looking for is called “Folk Religion” which means it’s not organized beyond what local groups chose to believe are the “rules.” Without more details, anything below might fit.

Animism is a starting point, in which you believe that everything has a spirit or is otherwise alive in a spiritual dimension. There aren’t gods, per se, but elemental forces are higher forces that are semi-sentient. So, for example, the Sun would be alive, Earth would be alive, the elemental force of water is alive, and each has some sort of sentience, but it’s sort of too high to directly talk with people, but you sort of communicate with feelings.

Shamanism is animism with more nuance. Gods, demigods, demiurges and the like exist - basically there are non-human, non-corporeal entities that operate in a spiritual realm, as do humans, so a shaman does negotiation as a middle-man because they have learned and been trained to be able to operate in both our realm and theirs. While not an organized religion, most forms of shamanism have similar rules and standards. Which is surprising considering that many cultures developed shamanism independently of each other.

As a sort of more detailed step towards specificity, you then have specific things like Native American traditional religion, Shintoism, many African traditional religions, Druidism and European pagan traditions, modern wiccan or other witchcraft-oriented beliefs, where local gods and spirits abound and are deserving of worship and veneration from everyone, not just having the shaman interceding on your behalf.

Slightly more organized, but not really, are polytheistic religions. Hinduism, Hellenism, the Roman Pantheon of gods, etc. Westerners think of these as “organized” but they really weren’t/aren’t in the way that we typically think. There was no main “Church of Zeus” and then after worshiping him, you go to Athena or Nike. A person and household had their god and they gave sacrifices, then also did the same for other gods if they needed their help. It was very ad-hoc, and sort of interesting, as the Greeks and Romans went around the ancient world meeting other cultures, they would find another polytheistic religion and not say “No, our god of war is Ares, and she’s stronger than your god of war.” They assumed that the gods were the same globally, and it was just the names that changed. So more like “Oh, you call the god of war Kartikeya? Cool, we call him Ares. You know him, too, awesome.” So the dogma is actually quite light.

Honorable mention for Taoism and Buddhism, which both can incorporate varying levels of animistic and shamanic beliefs, plus gods as higher beings that are out there, but not as high as every human’s inner Buddha form (if that makes sense). However, as philosophies-cum-religions go, there’s much more dogma and convention in play in some versions. However, both are Gnostic, in that personal experience plays a role in shaping a personal dogma, rather than having someone shout rules at you from a pulpit. I’m not familiar with the Taoist angle there, so I may be wrong about that to some degree.

There are subsets of monotheistic (primarily Abrahamic) religions that are mystical and are less dogmatic. Sufis or Kabalists or Christian Mystics. They sort of do their own thing, and typically are seen as maybe heretical, maybe not, by the mainstream elements of the same religion. This crosses over the last line of what you mentioned about dogma, but worth mentioning.

Finally, the Church of the Flying Spaghetti Monster is just about anything you want it to be, and there’s also a Church of the Invisible Pink Unicorn.