My BMW is a 2007 328I Coupe. Car needs new tires and has OEM 17 x 8 wheels (224/45 17 Tires). I would like to go with a larger tire, possibly staggered...What’s the largest size(s) tires can I put on these wheel?
The 8" cross section is the limiting factor, can't really go wider in tire cross-section than what you have. Look on craigslist for some 18" wheels in the 18x8 and 18x8.5 - make sure that they're from an E90 or F30 so they'll fit correct. Use 225/40R18 front and 255/35R18 rear.
The constraint for plus-sized fitment for wheels and tires is primarily safety-related - you want to stay within 3% or less of stock overall diameter. Despite the negative performance effects of a larger o.d., they should be relatively negligible if you stay within 3%. https://www.1010tires.com/About/Wheel-Plus-Sizing/ Most you can do on your stock 17x8 rims without going 3% over stock o.d. is 45 aspect ratio in 235 or 245 widths, so 245/45-17 or 235/45-17.
Personally, I wouldn't do a staggered setup or a tire that has a larger o.d. than stock. As I mentioned in my reply to your post in the E90 sub-forum, I'd do a square setup for more balanced, neutral handling, and the ability to rotate tires around the car, even out their wear, and maximize their usable life all together. A staggered setup, the rears as the powered wheels wear out way before the fronts, and can't be rotated but side-to-side, meaning they have to be dismounted and flipped on the rims to even out their wear, and the inboard shoulder and 1/3 to 1/2 will be down to wearbars with tread left in the middle. A square setup, the middle wears when they're moved to the front since stock front camber is zero degrees, so as long as you swap fronts to rear at regular intervals, the wear will even out; and you can replace all tires at once, and not have the disparity of mechanical grip between fronts and rears from having one end or the other wear out-of-sync with the other end, needing replaced, and end up with the rears on new tires but the fronts having a long way to go (for instance). 245/40-17 is closest to stock overall diameter, only .3" less, a negligible difference; 235/40 is probably an easier fit on the stock 8" rims, the size comparer I previously linked shows those are .6" less o.d. from stock. Unless you need some additional ground clearance for some reason, there's little performance benefit to a tire setup that's larger o.d. than stock; it's an effective gearing change which means for any given gear you'll be at lower rpms, so the car will accelerate less-quickly comparatively. You would gain some protection from bent rims/blowouts with the additional sidewall, and presumably some ride comfort. Tire turn-in response would be somewhat less-immediate, but probably not particularly noticeable. Cruising mpg. should be slightly improved, but also probably not particularly noticeable. A taller 245 probably would mount on the stock rim a little easier, more sidewall means more flex. Since there is some variation of tread width depending on tire and manufacturer, it might be worth having a conversation with someone from the Tire Rack about whatever tire you consider going with.
