Windows' own "small fonts" at 4 or 5pt size seems to come out quite well, though you might have to pick and choose what characters look OK and which don't. If it's actually rendered properly (you may have to turn off Cleartype, or even use Paint in 1-bit monochrome colour more) then it should come out looking a bit like Berkelium above, which isn't far off a minimum-size hand-tuned face I used to use (either zoomed in and hand-pixeling, or with lots of high speed copy and paste) for putting very small text onto low resolution images.
There's so little space for conveying information in that tiny 5x5 (or 5xless) grid that you have to get a bit creative with letter shapes in some cases, like the "e" in Berkelium (it also has interesting lowercase "a" and "s" shapes IIRC), for each one to be both readable and identifiably unique. And it may well be useful to use proportional width, so you can save on pixels with thinner letters in order to spend them on those where extra width is necessary.
As someone hinted at in passing above, the 6x6 system font used on the Atari ST (for drive and uppercase file labels in the main, with 8x8 or 8x16 used elsewhere, though not exclusively) may also be worth your attention. It's essentially 5x5 the same way that the others are essentially 7x7 and 7x15, as they have vertical and horizontal spacing built in for all but the deepest descenders and box-drawing characters. Notably ISTR the e's and a's lacked openings in their loops, but they were still recognisable (and as demonstrated, if you end up modifying them into a custom typeface, you can fix that).
Other than that the smallest you tend to get in common use is 6x6 or 6x8, with or without surrounding space, or 5x7 on some dot matrix LCDs. It's hard to make a decent font with any kind of character to it rather than pure robotic functionality with fewer pixels (indeed the other examples given above feel rather familiar, as they're either identical or very similar to ones I've seen used regularly on low-rez mobile phone LCDs back in the late 90s/early 2000s). Similar to how it can be difficult to make letters (and certainly to have a choice over using upper or lower case) with 7-segment numeric LCDs, and it's not much better with 9 or 11(+) segment. And 5x5 pixel is essentially 25-segment, except really it's more like a slightly more flexible kind of 11 or 15 seg. That's the kind of constraint you're working with here.
If you can stand it, just plonk the pixels by hand... at least you have ultimate control over the hinting/kerning, then, which as mentioned by others is probably going to be your biggest issue unless your font engine is specifically tuned for, or can have an option set to tune for pixel based rendering of small typefaces, rather than assuming everything is infinite-resolution vector and of a decent size. (if you scaled up a tiny pixel font to a normal size and morphed full-size type to conform to its critical boundaries, it would look quite strange, the same as trying to cram a large face down to minimal pixels - your minimum spacing granularity is ENORMOUS vs the actual glyphs)