An alternative idea that allows lifting to effectively unlimited height (even taking material strengths into account) is a chain drive bucket system.
The idea is as simple as buckets of water on ropes (the buckets can be as big as barrels).
If the rope would be too long; you can build in a relay system (basically gearing); so (for example) your water wheel is designed to turn an axle, the axle turns a rope with knots in it (to prevent slippage), the rope has permanently attached wooden buckets about the size of barrels. seal them with pitch on the inside, it is waterproof and lasts a long time.
Say our first stage is going to lift the water 50 feet (the height of a modern 4 story building).
These are lifted up, full of water, at the top of the wheel the rope goes around the top axle; which it is also turning. The buckets are guided by simple barriers to dump their water into a chute (also pitch sealed). Either you are now done, or you go on to the next stage: That same top axle, on the other end, has the same setup: Another chain drive and buckets it carries upward to be eventually dumped in another chute.
That is the Stage 2 Lift: Stage 1 collects water from the river (and is turned by the energy of the river). At the top of Stage 1, the chute (and timing) are designed to automatically dump the Stage 1 bucket into a chute, that fills an empty bucket of Stage 2, which is designed to be "going by" at that time.
Stage 2 rises another 50 feet, and dumps its bucket into a chute: Which could lead to a Stage 3 lift.
There is the strain on the rope of turning all the Stages at the same time, so there is a limit. But the rope does not have to be the breaking factor, you can just use 2 or 3 ropes if the strain is too much. The breaking factor is now strong you can make the water wheel before it breaks, or a tree trunk axle breaks, under the strain. Also, fewer buckets can be used if need be; each stage only needs to be carrying one at a time in order for this to work.
This scheme bypasses the need for a super-strong ropes that might break.
The gear that holds the rope can have a "string of beads" design; imagine a rope with regularly spaced knots: It casts a shadow that looks like a string of beads; basically circles connected by a thick line. That same shape should be in the trunk of a large tree or on a wheel, so the rope, with knots in it, has knots that fall into circles, and the rope between the knots falls into thinner channel. It is way to make a chain drive without any chain or gears. (You want the chain drive for its accuracy in a clockwork mechanism like this).
Stages above the first would need platforms built into the side of the cliff; but those do not have to be too extensive; people don't have to live there or anything. You can "dig in" to put some tree trunks about 20 feet into into the side of the cliff; they will take the strain of holding the axles in place and supporting the few chutes you need, along with some flat space for a maintenance worker to make any repairs (or rope replacement) that might be needed.