Why don't we have mipmapping on the web? I've never heard anyone suggest it before, but it seems like a good solution to this problem and all the other pixel ratio kinds of problems.
If you mean ability to supply several, potentially hand-crafted images for different sizes — that's exactly what .ico already offers.
If you stick to the precise definition of mippmapping, where they're exact downscalings of one texture, that solves a different problem — efficient downscaling, only sampling O(1) texture pixels per displayed pixel.
On the web CPU isn't the problem; there are 2 other reasons so serve several sizes:
(1) Tiny images (like favicons) don't scale well, you might want manual intervention.
(2) Too large images waste bandwidth => you want negotiation, especially to transmit less to mobile devices.
.ico was a quite elegant solution to (1) actually. I guess all the mess comes from icon resolutions growing enough to turn the problem into (2)...
There several efforts to solve this generally — google "Responsive Images", "srcset".
I'd think progressive JPEG and PNG could have solved (2) well — just stop downloading when you had enough quality — but many people say that's not enough :-(
