One thing to watch out for is the MCP1624 uses PFM at low power.  The low switching frequency can mess up analog/audio circuit.  MCP1263 would be better for those circuit as it has fixed frequency, but at the expense of lower efficiency.  The high PWM frequency is easy to filter.

I learnt this lesson the hard way in my visual impair project a few years back.

I saw the two versions in the datasheet.  That's a good tip - I chose th 1624 just based on the efficiency at low currents.

Analog circuits might have a problem with the wide range of supply voltages they'd might see with this kind of supply, too.

I am try solve this problem on my supercapacitor watch without success almost all stored energy in stanby is consumed with step up converter. I think some time about attiny43u. I make one board with max1724 which have 1,5uA qc but your idea is very effective.

This is interesting project. Just by coincidence, I made research about single cell MCUs. My idea was to make particular device that can run of single Ag cell, but eventually I gave up.

I haven't found that much. There are offerings from Microchip and Silabs, with internal step-up circuitry (add inductor and there you go); both with not that much inspiring parameters. Both being 8-bitters, no 32-bit offerings; also both manufacturers do have just a few models, with not much of scalability. There is MSP430L092 that runs "natively" from single cell, but has no internal FLASH, you have to add external one via SPI. Sounds acceptable, but good luck finding anything (FLASH, EEPROM, SPI RAM) that runs from single cell, so TI's recommendation is to make bitbanged step-up to power the FLASH. A bit of meh solution.

Also, I found out there are next to none user output devices. Finding 1,5V LCD is hard, LEDs do not work much here and filament lightbulbs may work, but do not make much sense here.

On the other hand, for decades I'm surrounded by cheap electronics that run for ages on single LR44 or similar cell. Calculators, alarm clocks, thermometers, you name it. Brainbox is hidden under drop of black snot on PCB, with no apparent inductors, so it runs "natively" on 1,5V, including program storage. Is ROM key here?

The core voltage on modern CPUs is in the single-cell range, down to 1.2 or so for slow clocks.  Plus all the simpler devices you mention.  I doubt it's a technical issue, really. You can buy modern, fast logic gates (e.g. 74AXP) that work down to 0.7V:

https://www.digikey.com/product-detail/en/nexperia-usa-inc/74AXP1G08GMH/1727-1837-2-ND/4843077

I think it's a market problem.  Li batteries are everywhere, and compared with these, most primary cells are expensive and huge on a per-Coulomb basis.  Add in the lack of available displays, and maybe a low-voltage MCU just doesn't make sense.

Yes, probably low market drive.

I noticed the low-voltage logic gates when browsing for low-voltage MCUs and really thought of replicating my 4-bit CPU with those. So I'd finally have true single cell CPU :-)