I like Bezos's suggestion of having shorter patents for software. 3-5 years is enough to protect innovators, but not long enough to stifle further innovation, and probably not long enough for patent trolls to purchase and then sue people.
I can write an entire book under 1 copyright, easily keeping it going for a decade with your regime. But a photographer produces hundreds or thousands of copyrighted works a year.
You might try doing your regime with the number of claims, and the company can decide how many to keep fresh, but it then leads to weird edge-cases like keeping a few things secret for now to patent later -- which is the opposite of what the patent system intends. Patents encourage people to publish their inventions for others to study, in exchange for limited exclusivity to market and/or use their invention.
Keeping things secret now to patent them later is a problem, but no worse than it is now, I think. The same incentives apply to the exponential scheme as to the current hard limit. You'd keep the requirement to file within a certain time of the first public reveal, so you'd have to move quickly if you wanted to patent anything you were actually going to use.