There's "Day Million" and The Age of the Pussyfoot, both by Frederik Pohl, and even they might be more like utopias rather than adequately weird.

"Day Million" is a very short story, with a narration which puts emphasis on how much people like living in that world even though it would make little sense to a contemporary reader. Unfortunately, the most vivid detail is a spoiler. N pbhcyr (obgu bs jubz jbhyq frrz bqq, gubhtu bar'f pyrneyl znyr naq gur bgure'f pyrneyl srznyr ner nggenpgrq gb rnpu bgure-- gurl unir n tbbq gvzr bapr, naq gura qb jung'f abezny va gung phygher-- rkpunatr vqragvgl gncrf

Silverberg's The World Inside might count.

It a description of how people might be pretty happy living in a maximum population world.

R.A. Lafferty's Slow Tuesday Night is  a weirdtopia, and so is his Primary Education of the Camiroi. Unfortunately, his "Polity and Custom among the Camiroi" is incomplete at google books, but I recommend getting a paper copy of Nine Hundred Grandmothers-- the collection has some of his best work.

gwern supplied the link for Polity and Custom:

No assembly on Camiroi for purposes of entertainment may exceed thirty-nine persons. No more than this number may witness any spectacle or drama, or hear a musical presentation, or watch a sporting event. This is to prevent the citizens from becoming mere spectators rather than originators or partakers. Similarly, no writing -- other than certain rare official promulgations -- may be issued in more than thirty-nine copies in one month. This, it seems to us, is a conservative ruling to prevent popular enthusiasms. A father of a family who twice in five years appeals to specialists for such things as simple surgery for members of his household, or legal or financial or medical advice, or any such things as he himself should be capable of doing, shall lose his citizenship. It seems to us that this ruling obstructs the Camiroi from the full fruits of progress and research.

"Slow Tuesday Night" is a whimsy about people who've had a mental stutter removed-- they live so fast that they can have three careers in eight hours. "Primary Education Among the Camiroi" is about a culture which develops maximum intelligence and self-reliance, at the cost of a few of the children being killed. It teaches slow reading (reading slowly enough that everything is remembered), and the world government course consists of governing a world (not a first aspect world) for three or four months.

"Winthrop Was Stubborn" by William Tenn-- a group of time travelers are trapped in the future because one of them doesn't want to go home. I only remember a little of it-- I think there was artificial living/moving food-- but the point of the story was to portray a society which was weird for a modern reader and delightful for its inhabitants.

Any other nominations?