Besides the regular hand consisting of four sets and a pair, nearly every style of mahjong also allows for some irregular hands. These are hands that are usually made of single tiles and pairs of tiles. Since these hands do not contain triplets or sequences, you must get at least 13 of the tiles yourself, but you are allowed to claim the last tile to go out. Zung Jung allows two irregular hands, and since both must always be hidden you are not allowed to claim points for a concealed hand.
10 Irregular Hands
10.1 Thirteen Terminals : 160
10.2 Seven Pairs : 30
10.1 Thirteen Terminals
Also called Thirteen Orphans, this pattern is recognized in most mahjong styles. There are 13 terminal and honor tiles: 1 and 9 of each of the three suits, plus four winds, plus three dragons. To score this pattern, your hand must have exactly one each of 12 of those, and a pair of the 13th. You do not need a 13-sided wait, so you could have a pair of green dragons, and need just a North wind to go out.
Most other styles score this as a limit hand, but it is by far the easiest of any pattern that scores a limit. To give you some idea of the odds, it's just a little tougher than getting a full flush with a concealed hand.
You have a pretty good shot at this if your starting hand has seven or eight of the 13 tiles. It's also somewhat easy to detect and defend against. It's also somewhat likely that the hand will become impossible because of a kong, or discard of all four of a tile you need. Fortunately, if your hand does become impossible, you often have plenty of safe tiles to throw away.
You cannot combine this with All Terminals and Honors by definition, nor any other pattern. You can combine it with the incidental bonuses such as last tile, last discard, or robbing a kong, which is especially fun.
10.2 Seven Pairs
Saved for last, but very easy to explain: In your 14 tiles, you have seven pairs of identical tiles. You are allowed to have sets that match each other. You are not allowed to claim a kong, because you'd then have a 15th (or more) tile and could no longer make exactly seven pairs.
It turns out this hand is more common to obtain than all simples plus concealed hand! Of course, that's ignoring how the game plays. Since this hand scores the same as four triplets, you can easily switch your tactics and go for that pattern, but that's pretty hard too.
This pattern cannot combine with any pattern that calls for triplets or sequences, but can combine with whole-hand patterns such as all simples, a flush, or all terminals and honors. You cannot combine with patterns like all honors, only because those other patterns are listed limit hands, but you can score all honors or all terminals with a seven pair hand.