Why not introduce a battery source to RAM to preserve its contents across boots?
Actually, your computer is already using battery-powered memory. Have you ever had to reset your BIOS settings when you couldn't boot the computer? There are two ways to do it: with the jumper or by removing the CMOS battery. That battery is used to power little amount of memory used to keep BIOS settings. It's not RAM, but it's close to what you're asking for - as you can see we're already using powered memory.
As of now we aren't using powered RAM because we have SSDs. Consumer-grade SSDs let you boot your OS in seconds and they are way cheaper than RAM. That's it when it comes to reading files from disk.
But SSDs are total game changers. So why weren't we using powered RAM before the SSD era? Because 20 GB (the size of my Windows folder) of HDD space costs $2.2 and 20 GB of DDR3 costs $220 (best offers from Newegg). If you can afford that, there are some options available.
I'm talking only about keeping files in RAM because with current hardware and software design, keeping state across reboots is impossible. Keltari has already posted that for example changing the graphics card could be destructive for the hardware or software. The same applies to performing disk modifications.
For example the same file in RAM and on hard disk could lose sync at some point if you take out the drive (or RAM) and mess around with some files). Checking if RAM files match disk files is as slow as simply reading from the disk, so it would defeat the purpose. You'd have to remember to reset RAM (like the CMOS battery thing) or create hardware systems that would do that, but they would require a power source too. Building such system is just hard, if not impossible.
And, finally, state is meant to be volatile - as the opposite to files, which should remain. That's how computers are designed. To change that, you'd have to rethink everything what we have come to since volatile RAM was first used in a computer.