I'm on my 7th chain this year, but I'm on a fixie. I think a road fixed gear, particularly if ridden in the rain and up lots of hills, puts far more demands on a chain than a geared bike, particularly a geared bike that sees little rain. Because you can't shift down, you're torquing a fixed gear out of the saddle at much higher forces than you'd be sitting and spinning. And in the rain, you get grit off the wet pavement grinding away the gear teeth.
I'm at just about 17,500 km for the year on the fixed gear bike, so I've been replacing the chain when I start hearing it make funny noises, or feels like it's "catching" under pressure, due primarily by its failure to mesh properly with the rear cog. This works out to between 2,000 and 3,000 km, depending on time of year. I even switch out cogs: a new or relatively unworn cog with a new chain, then a switch to a medium-worn cog at about 1,200 to 1,500 km, then to a worn cog once over 2,000 km (which I often throw out with the chain) when the chain is definitely stretched, always trying to match chain stretch with cog wear so that I avoid the "grinding" noise when I'm pedaling under pressure. But then, part of the problem could be that I use cheap cogs made of "soft" steel that deforms into "shark fins" after only 6,000 km or so. But I can't see spending more than $10 or $15 for a cog, and some of them run for $35!
Now contrast this with my track bike, used on a level surface (unless I'm going up the banking), indoors, where the chain will last for years!
With the road bike (and on the tandem), I try to avoid riding in the rain (that's what the fixie is for), and I replace the chain at about 3,000 km before it's had a chance to wear out the cassette. If there's any skipping with a new chain, I try to just replace the skipping cog, rather than replace the entire cassette, which I think is wasteful. Another reason to ride a fixed gear.
Luis