Edit: Re-reading this, particularly that the problem is users trying to tap for more info - any icon is likely to look like a button that can be pressed. Icons almost by definition are intended to attract attention and look like something functional: whereas you want something that looks like background.
So given that, I think it has to look like a joining line - background and structure, not foreground. How about something like
...but where the vertical lines (one for each day passed) are very thin and close together so there's not too big a gap?
The principle behind this suggestion is: to carry information but not look like a button or interaction point, it has to carry information while looking like part of the background structure - not foreground. It has to be something people glance past, not something people are drawn to look at directly.
Original answer: if it was okay for it to look like an icon (therefore, potentially like a button)...
Since you're specifically looking at days, there are two common conventions you can use:
- Something often used in films, TV, comics etc to signify passing days: crossing off days on a calendar, mixed with...
- ...that common, familiar type of calendar icon (taken from the Noun Project):
...but of course you need to make it clear that it's the passing of days that is the focus, not the calendar itself. You could draw a calendar so the calendar itself is clearly recognisable but background, and crossed off days are the foreground, something like this:
Depending on how much space you have, you could potentially have different icons for smaller and larger amounts of time passing:
...but don't get carried away! Keep it simple.