As a practical matter, laptop batteries are limited by FAA regulations: no more than 12v and no more than 100 watt-hours. [1] A laptop that runs off of battery power for a week probably must be engineered backward from the 100 watt-hour limitation. A week has 168 hours so a processor running continuously at one watt uses the entire battery budget in about four days without powering a screen or a network chip. [2]
Good luck.
[1]: https://www.faa.gov/about/office_org/headquarters_offices/as...
[2]: None of which means the idea isn't a viable Kickstarter.
Might run your laptop for a week: https://www.amazon.com/gp/product/B06W9FWFXD/ref=oh_aui_sear...