There's more than a few standard designs for planar and conical wideband antennas, with smooth, periodic, or log-periodic (self-similar) design.
There's actually a theorem that any structure you build, if it's self complementary and repeats forever, is exactly half Zo (377 / 2 ohms), at all frequencies. And all that resistance is radiation resistance, so, it's one heck of a wideband antenna.
BTW, these aren't to be made with wire -- you need wide, flat (or round) conductors to do this. I mean, you can use a mesh equivalent too, but that's way more work than cutting up foil or sheet metal!
Personally, I rather like the idea of a monstrously large ridged horn antenna. I doubt one is very practical for, say, even the upper half of the shortwave band (it'll be some meters wide!), and construction needs to be extremely accurate to work over a wide bandwidth (obviously, surfaces need to be within 1/4 wavelength at whatever the upper "still pretty okay" frequency should be). But they do look cool.
Easy, lower frequency, wide-ish designs include the bowtie and biconical dipole. Commonly seen in EMC test labs, for good reason.
Tim