ALL of the single engine Cessna's I've flown (150, 152, 172, 177, 182, 206 and 208) will climb with full flaps. My initial CFI had a habit of making you fly a pattern with full flaps. Field elevation there was 1350', it was May and June in northwest Arkansas so density altitude was up there a bit. The airplanes will do it, it just isn't a lot of fun.
The reduction to 30 degree max on the later 172s was in part due to the increase in certificated gross weight and the need to be able to climb with flaps out, part of the certification requirements IIRC.
****Edit. Part 23.77 requires the aircraft to "be able to maintain a steady gradient of climb of at least 3.3%" using takeoff power, flaps in the landing position.