Generally 1 1/8" is generally recommended. I see new houses built with tile right on top of the 3/4" subfloor all the time. Jump and down and give the bounce test. It should be ok. If not then consider some blocking in the joists. There is no absolute requirement for cement backerboard and it is usually used in a wet situation such as the bathroom when an underlayment is needed anyway. You have plenty of structural support now. If anything check to see any any additional screws will help hold it down more solid. Good luck.

Thanks for the response, bc. My biggest problem now is that the current (inadequate) 1/2" subfloor plywood extends under all the walls of the house...interior and exterior. The walls are log. I am considering using a sawzall to cut the current subfloor up to the wall, then add some blocking or joists underneath to support my new 3/4" T&G plywood subfloor.
Are there any complications with this approach? The weight of the walls will still rest on 1/2" residual 'strips' of plywood, and the floor will no longer be tied into the walls. I'm pretty sure it isn't easy to fit the new 3/4" plywood under the walls.

You kind of lost me on this and I can't visualize what you mean. I suggest getting a couple bids from tiling contractors and see what they say. Don't go cutting on the floor until someone can look at it and say OK. I'm not sure I see a need to cut the old floor.

OK, I've decided that I cannot remove my
1/2" plywood subfloor, but that I will lay down a substrate for the tile over it. However, my new dilemma is what to lay down. I'm considering 3/4" T&G plywood, then tile over that. Is this acceptable for tile? The other option could be Concrete Backerboard, but this only comes in 1/2" thickness maximum. I need to build the floor up at least 3/4", but I can't go to much higher than that. Does anyone know of any concrete backerboard thicker than 1/2", and would this thicker concrete board over the existing 1/2" plywood subfloor be an acceptable surface for tile?

You already have 1 1/4" with the 1/2" plywood and 3/4" particle board. That should be plenty. Unless the particle board got wet and is damaged which would then need to be replaced, I'm not sure what your concern is. Make sure the particle board is screwed down.

I think it is inadvisable to tile over the particle board.