I'll add one more that I have run across. It's a revenue/savings share agreement, where the company writing the software gets an X% of the revenue (or savings) generated by the project.
One problem is that clients hate it, even when success was very measurable. I suspect due to our natural tendency for zero-sum thinking. Note that usually the projects were limited to ones where there was a directly measurable outcome to drive the metrics for the payments.
Wierdly enough you would think that such a business could pivot to developing its own product. Why is it that custom software dev houses hardly ever manage to pivot to developing a successful product?