Like most people, my favorite Asher book keeps changing. Scrabbling through my sketchy records it looks like I probably first read "Owner Space" (2008-09-10) and actually didn't much like it. Then I read "Alien Archeology" (2008-09-20 - which I've elsewhere mistakenly said was first) which I liked better and that prompted me to pick up Gridlinked (2008-09-30 - seem to have a thing with multiples of ten days) but I can't remember if I read it or The Engineer Reconditioned first. I'm almost positive I read it first, before reading the collection by 2009-05-18. The novel was pretty good and made me think I might like this Asher guy but wasn't so great that I was completely convinced at that point. But the collection made me pretty sure and, at that point, it was my favorite.
Then I spent a long time piling up Asher books, trying to get the right ones in the right order and I'm finally getting around to them. Since Christmas I've read Cowl, The Line of Polity, Brass Man, and Prador Moon. I read Cowl while I was sick and I often dislike time travel and this book wasn't perfect (and neither was I) but it still seemed to have at least really good parts. Then, with its excellent alien ecology, interestingly vicious sociology and cool characters, The Line of Polity became my new favorite (or at least tied the collection). While still spiffy, I actually didn't like Brass Man as much but I'm still looking forward to #4 and #5 of course. And now I've read Prador Moon and that bit of concentrated explosive is now my favorite Asher.
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The topic's "favorite" but there's a lot of discussion about "where to start" and I'd say that, of the six I've read, The Engineer Reconditioned (a good array of what Asher can do), Gridlinked (his first major novel and first (in publication order) Cormac novel and first (in internal main series order - though there is a prequel)) is another decent first. And, arguably, Prador Moon (a short concentrated dose and internally the first Polity book - of which Cormac is a part) would be the best starting place.
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Re: Banks - I may yet be a Banks fan (only read three and still not sure - actually doubting now) so it's possible but, in many ways, Asher and Banks share some of the same science fictional furniture but seem antithetical and I'd imagine there's a lot of discrepancy in their fan bases, too. Not everything's black and white in Asher and not quite everything is a perfect gray in Banks but Asher has a much more, um, vigorous depiction of settling differences and a clearer picture of who should win. And Asher seems to depict a much less sybaritic universe. Banks' narrative voice seems to deplore violence but admit its necessity while Asher's characters don't really glorify it but are much readier to take up arms. And there's a difference there: Banks is very much a "narrator" guy while Asher lets his characters' actions do a lot more of the talking. And Asher himself (above) is correct as far as I'm concerned in that I've read Banks' first three and prefer Consider Phlebas by a good margin. (The Player of Games trails by a good margin, with Use of Weapons somewhere between.)