22" x 14" x 9" is used by American Airlines, Delta, United, Virgin Atlantic and probably quite a few more.
Ryanair is 55 cm x 40 cm x 20 cm that is 21.5 x 15.7 x 7.8 Inches. There are other low costs using this size. Note that many airlines use 55 x 40 x 23 but according to http://travel.stackexchange.com/a/30181/4188 the sizer is only 21.2cm so I would not expect a 23cm bag to work out with Ryanair. Of course unless you have a hard shell or packed your bag to the gills some squeezing might allow for a 23-thick-advertised bag to fit the sizer.
Therefore 21.5" x 14" x 7.8" / 55 cm x 35 cm x 20 cm is the biggest bag you can safely carry. This is the IATA recommendation as well per this 2015 June BBC article. The closest non-wheeled (because that's just extra weight which you don't need when you have a severe weight limit often anyways) I am aware of is the Magma Digi Control XL backpack is 54 x 35 x 20 cm which I just bought exactly because of this and also because I have a 19.5" monitor to carry on and it happens to fit into its laptop pocket which is pretty rare and it has a second laptop pocket which pushes it into unicorn territory. Has its own drawbacks: it doesn't open like a suitcase or a "convertible" backpack and also there is no size handle.
You might be able to get away with slightly bigger: the main US airlines' sizers are actually 23" x 15" x 10" United Delta so we can conclude that 15" instead of 14" is doable which increases your maximum bag to 21.5" x 15 " x 7.8" / 55cm x 38cm x 20cm. This is the size Emirates actually allows.
So if you want to go nuts, buy a 22 x 15 x 9 / 56cm x 38cm x 23cm duffel style bag if you can find one, pack lightly (8kg is not unheard of as a limit, 10kg is common) and prepare to squeeze. Perhaps you want the Red Oxx Medium Aviator Bag and a few of their luggage compression straps that you set up to cinch it to a width of 9" -- the other two sizes are just what you need, I have the Extra Small One, I find the price surprisingly low given the building quality. Of course, there's no structure but that's what you want from it. Get packing cubes :)