Need beam for a 22-foot span. It's a carport. What to use?

I'd like to build a carport. It will basically just be a shed roof extending off the front of my garage. I'll need a beam, as thin as possible, to span 22 feet from side to side.

It looks to me like steel is the only thing that could easily do that, but the cost is high, and how thick does it have to be? I priced out some 6" i-beams. $500 each. How do I attach wood to those? Can't exactly nail into them.

That brings me to LVLs and such. Less pricey, but very thick. I suppose I could double up on a couple of thinner ones (like 6") but they don't seem to be rated for exterior use, and I'm pretty sure a carport would be considered exterior use, even though they'd be under a roof.

Any guidance?