Beaglebone (from Texas Instruments) is more open, but still not as open as you'd probably like. Still, its a better balance than what Rasp. Pi organization has (more documents are available on AM335x, open-hardware for Beaglebone Green and reference designs, full chip specifications and the like). Beaglebone isn't really "more expensive", as much as its just "lower specs at the same price" though.
The "most open" are MPU chips and their associated "System on Module" boards. This isn't quite a SBC, but its easier to use than a BGA. These SoMs are very poor from price/performance perspectives, but instead serve as reference designs and/or prototypes to the $8 or $9 chips. The overall expectation is that you're "supposed" to be building your own PCBs eventually, so the SoM are kind of just a prototyping aid.
Most SoM provide 100+ pins from the chip as well, meaning you absolutely have to build a PCB to use them. However, 2-layer boards solder very easily to a SoM with castilliated edges (even with a hand-soldering iron)... albeit with a bit of flux and technique and practice. Its just the easiest way to deliver the most-pins of customization in the smallest space. So a relative beginer should be able to boot an SoM. The most difficult routing and Power-Delivery-Network details are already solved on an SoM, you just gotta apply power and build out the final interfaces / connectors.
Take the ATSAMA5D27-SOM1, 104-pins in a 40x40mm form factor. $50 from Mouser for 500Mhz and 128MB RAM (though fully open source and fully documented at linux4sam, and processor manual, U-boot process and everything). But the underlying SiP (MPU + DDR2 RAM) is like $15... while the MPU alone is like $8 and 128MB of DDR2 RAM is only like $3.50 in practice. Since in mass-production, you'd probably have a custom PCB anyway, that's the most expected use case. https://www.linux4sam.org/bin/view/Linux4SAM . I'd say that Microchip / Atmel's MPUs seem to be the best documented that I've found, but are unfortunately the lowest specs. Still, they also have some of the lowest power-consumption (like 200 mW or something), so really they're in a low-power class of their own. Still Linux though.
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STM32MP1 is the MPU from ST Micro. Like the Microchip SAM-MPU series, the STM32MP1 is available in SOM, SiP, and "raw" MPU form. Except the SOMs are like $100+, the SiP is like $50+, while MPU is $10ish.
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I know NXP has a huge line of MPUs. I haven't researched them yet though.
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I think all the hardware designers at this level just "assume" that their customers, if they care about "open source", are probably making their own PCBs.
If someone "just" wants a SBC (like the Rasp. Pi), there's not much point in publishing a ton of documents. People can just boot the Rasp. Pi and start messing with Linux.
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I got no experience with this yet. I'm just curious and am thinking of a simple MPU layout project ever since I discovered that OSHPark has 6-layer boards and KiCAD supports BGAs in practice. Overall, these lower-power lower-end MPUs fill a different niche than a Rasp. Pi ever would. But I feel like there's enough overlap that these might scratch your "open source" and "fully documented" itches.