I'm close to finishing my 5" x 1/2" engineered wood flooring. I noticed that only on my longest wall, I accidentally left more than the 1/2" expansion gap that us around all of the other walls. Probably close to 3/4" - 1" gap on this wall. My baseboards are 5 1/4 wide and about 9/16" thick. I really do not want a shoe molding or quarter round because of this one wall. What options do I have? I would love to just have my new base boards sit on top of the wood without any molding.......

Well, if that longest wall was fitted with a "rip" (partial board), you could remove it and re-cut a new piece(s) to fit closer to the wall.
To be entirely safe, you should also undercut the drywall to allow greater expansion space.
If the last board is a full one, you will need to cut a narrow rip and add it to the space you have. Downside to this is that the line of that partial board will be immediately in front of the bottom edge of the baseboard. Not the most attractive thing, but do-able nevertheless.