Cryptic puzzles are often referred to as "cryptic crosswords," although many of them, especially the most challenging ones, are structured in a non-crossword grid.
Such puzzles seem to be less common in the U.S. than in some other countries. When I was in England it seemed that many daily papers had cryptic puzzles, and I have subsequently found them online in the Sydney Morning Herald, for example. But in the U.S. they seem to be relatively uncommon or unappreciated.
Some of the best cryptic puzzles I've encountered have been those in The Atlantic and in Harper's Magazine (the latter I believe only available in the print edition). Puzzles in both publications often require clue answers to be transformed in some way before entering them in nontraditional grids. (Both magazines also offer the advantage of being structured almost as aggregator-type blogs in terms of the wide variety of material they offer for reading).
Perhaps even more challenging than the Atlantic and Harper's puzzles are the ones created by John de Cuevas for Harvard Magazine and for the Puzzlecrypt. The standard grid, shown above, resembles that of a diagramless crossword. The answers to the 20-25 across clues are entered sequentially top-to-bottom, left-to-right, and the answers to the down clues entered similarly in sequence, but one isn't told how long each answer is. Furthermore, some permutation is usually required of the answer (typically the subtraction of several letters) before it is entered. Plan to set aside several hours when you tackle these puzzles.
As examples and teasers, here are portions of some clues (with answers) I've found in various cryptic puzzles:
"99/99 of a figure" = iconic [IC on IC]
"successor to Brand X" = brandy
"doubly evil sailor" = Sinbad
"1999 + 1" = immunity [IMM][unity]
"red birds" = regrets [r/egrets]
"N.D. terminal" = end [en]d
"number that is 2/3 of six" = nine [ix]
By the way, if anyone out there knows of additional online sites for incredibly challenging, brain-exercising, time-consuming cryptic puzzles, please... don't tell me.
