Best type of wood for shimming up a new plywood floor?

[BasicWoodStuff]

I am laying a new 3/4" plywood floor over an uneven wood floor, the room is going to be an office for me, where I want to roll around on a super strong, non-squeaky floor that is flat and extremely stable, that would remind me of my previous floor which was concrete with a thin carpet on it. 

My plan is to cut strips of wood to make up the difference necessary to create new flat surfaces to screw the floor to, just above the existing 2x6's that the current floor is fastened to, without removing the old wood.

The floor is far more wavy than it appears, and is slanted, so in one corner it's at zero, but the second corner it's 1" lower than corner one, and at corner 3 it's almost 2.5" lower.   The house has settled for the past 80 years, and won't be settling anymore (the supporting beams have been replaced to steel and concrete bases over the years) so now what's there is solid, and not going to move any more, it's just not straight/flat. I plan to continue this new plywood floor into the adjoining room, too.

So my question is this: I plan to use Liquid Nails to ensure a solid and quiet bond, but what kind of wood would be the most strong with the least amount of "give" to use as the make-up wood?    Is plywood scraps the best for this, or is any kind of 2x4 or 2x6 cut into the right strips as good as anything, if I use a bonding agent like liquid nails to use along with the screws?

I don't care about overkill (I prefer it- I'm going to be rolling around on this floor 16 hours a day every day in my office chair) so if anyone has any tips, I'd appreciate your input. I'm not an experienced wood worker but did work a lot of years as a laborer before my professional career, so between that and YouTube I plan to figure it out as I go. But the types of wood and the properties and what would work the best is the type of thing only an experienced person would know...  the green lines are a general idea how the strips would go, exactly above the existing support boards/beams (not sure of the term).  

Thanks again.

[Snipe Hunter]

What I would probably do it cut out and remove the planks. Sister and level new 2x4 nailers (nailed/screwed and glued) along the joists and nail and glue down a new ply or OSB subfloor. That way you aren't building up the height of the floor any more than necessary. I honestly think it would be much easier and you'd wind up with a better product with no squeaks.

If you want to do it over the existing planks, you could do it this way with sleepers. I would probably make my rips on a table saw though:

[BasicWoodStuff]

What I would probably do is...  

Thank you!   That's the idea of what I'm looking to do, and you're right- for as many as I'll do (and for other projects) a table saw will be a worthwhile investment.  You can find some pretty nice ones for $150 and the effort of picking it up on FB Marketplace and the like.

So those "sleeper strips" .... he just used a regular 2x4 board to make them, that's good enough?  There's no wood that is extra rigid, or expands less, or whatever?   Just whatever wood I can get from regular 2x4 or 2x6 boards are good enough?

[Snipe Hunter]

Thank you!   That's the idea of what I'm looking to do, and you're right- for as many as I'll do (and for other projects) a table saw will be a worthwhile investment.  You can find some pretty nice ones for $150 and the effort of picking it up on FB Marketplace and the like.

So those "sleeper strips" .... he just used a regular 2x4 board to make them, that's good enough?  There's no wood that is extra rigid, or expands less, or whatever?   Just whatever wood I can get from regular 2x4 or 2x6 boards are good enough?

Yes. Just 2x4s. Try to find straight ones. You could do this with a circular saw also. Nice sharp carbide toothed blade. Ripping gets dangerous with a cheap or dull blade. At the very least, you'll get a lot of smoke.

[crokett™]

I was going to suggest what Neil already has. It will be much easier to tear up the old subfloor and use 2x4s screwed and glued to the existing joists to level everything out than it will be to rip sleepers to the correct dimensions. Said more precisely, it will be easier to tear up the existing and use 2x4s to level the joists and have it be level and accurate than it will be to rip sleepers. Level everything to the highest point.