300 DPI is a rounded up figure for lithographic printing, where they should be multiplying the LPI of the device by 1.41 to get the DPI - that is if the imaging device has a LPI (lines per inch) of 150 then the DPI for an image should be approx 211.5 DPI.
Most people multiply the LPI x 2 to get the DPI - which is errouneous - hence multiplying 150LPI x 2 = 300 DPI.
But certainly if the device is 175 LPI * 1.5 = 262.5 or if it's 200LPI*x1.5 - 300
So 300 is a safe bet for a lot of lithographic printing.
Different lithogrpahic purposes use different LPIs - newspapers use about 80 - 120 LPI as the paper is pourous and can't absorb too much ink - if they ran at 150LPI on that type of paper the ink would never dry into the paper it would sit on top of it.
For newspapers you'd use between 120-180 DPI for your images.
With Billboards the equation is completely different - it's based on viewing distance!
Certainly for a bill board 300 PPI is way too much - you could get away with 40 PPI or even 72 PPI or even 150 PPI
Your print provider should be giving you details on what the size and resolution of the images should be before going to print.
However - when I'm sending large format files to the Printers for the purpose of large signs I turn off Compression altogether!
There's really no need to compress them - there's no reason to Downsample them.
You get a HUGE file - but with services like "we transfer" and "dropbox" "drive" and other cloud based file transfer system sending and receiivng large files is a cinch.
For billboards I think you're really talking about pixels per foot or per meter, rahter than per inch, but that's at print size. Without knowing how big the ID file is and how much it's going to be scaled I wouldn't want to give a figure for resolution in the file.