What is the cheapest ARM processor with an MMU so I don't have to use uCLinux? Bonus for non-bga and integrated RAM/Flash
The Atmel AT91SAM9260B comes in a LQFP package and may be a little bit lower in price than the SAM9G20. It has a MMU and the same peripherals as the SAM9G20 but runs at 200 MHz rather than a max of 400 MHz. Pay attention to the details in the Atmel Application note Schematic Checklists and you will be successful.
You could also try the AT91SAM9XE512 with flash on-chip. But 512KB isn't going to be enough even for uCLinux, so the most cost effective thing to do is use the flash-less SAM9G20 or SAM9260B with external flash and SDRAM.
I think the answer is chips in the ARM926EJ-S family, like the AT91SAM9G20. The G20 costs around $10 in relatively low quantity, and it has an MMU. Unfortunately, it's only available in a BGA package, and you'll need external memory.
The good news is that the ball pitch on the G20 is 0.5 mm (edit: there's a 0.8 mm pitch version too), which is just within the limit of what most assembly houses will do without a surcharge. If you go to an ARM Cortex A8, like the OMAP35xx, you have to deal with 0.4 mm ball pitch.
There are at least 5 companies that are making SBCs using the G20. Here are a few:
(I'm actually working on an open hardware board based around this chip, but it won't be ready for a few months at least. But that at least explains why I have all this crap in my head!)
The AT91RM9200 used to be a popular choice (ARM9 + MMU). It's non-BGA, but doesn't have integrated RAM/Flash.
The imx233 freescale processor costing around 10$ and is available in 128LQFP package
There are two another choices there:
1. Vybrid VF3xx ARM Cortex-A5 family from Freescale (now NXP), main features: 266MHz, 1.5MB on-chip SRAM, 2x Ethernet + L2 switch, LQFP176.
2. RZ/A1 ARM Cortex-A9 family from Renesas, they have few ICs with LQFP package (176, 208, 256 pins), and large on-chip SRAM (3MB-10MB), 400MHz.
You can get it from DigiKey for example.