My SR Laprade single-bolt seatpost doesn't grip the saddle rails tightly enough (saddle gradually slides back), even when I crank the bolt so tight I'm afraid of breaking it. Any remedies?

Is everything greased up? Threads, contact at the bolt head, the pivot (but not the rail clamping surfaces). I recall the "nut" sits in a round "groove". Try greasing or oiling that too. If yes to all of those and you still have issues, consider finding another Laprade or another post. Thousands of Laprades were made. Tens of thousands? Hundreds of thousands? A lot. In general, they were posts that while not elegant, worked reliably. Hence bike manufacturers spec'ed them regularly. But they are not new! Yours may have suffered abuse at some time. Finding one cheap in OK condition shouldn't be very difficult, either on-line or at a coop or bike shop that has used bikes.

Two strips of emery cloth between the rails and the clamp.

One I got on a used bike had the top plate bent so it wasn't contacting the seat rails well even with lots of torque. Took me a while to see it but after looking at it closely it was obvious.
So just take a look before you add another 90 degrees to the bolt....

I’ve seen this issue come up in discussion a few times and have experienced the same problem with the seat post, myself. Working from memory, seems the post is only rated for riders up to 220 pounds or so. At that weight or heavier and you can experience the slippage no matter how tight the clamp is.

I am 190 lbs, do not know the history of the post. It looks fine. Greased the bolt and put a couple strips of sandpaper in the grooves, we'll see how it works.

kroozer-

If your Laprade has the ridged aluminum cradle, you might want to take a triangular file to the cradle and the post head serrations. One of my SRs had worn the ridges down and did not seat properly allowing movement.