Most Valuable HKOS Hands

In our Hong Kong Old Style mahjong rules, scoring more than 13 faan will always be worth the limit of points, but because there are a finite amount of yaku, there is a theoretical upper limit of how many faan a hand can score.

FAAN FROM HONOR TILES
Your hand cannot be entirely composed of honor tiles, nor can it have pungs of all three dragons, or it would score a natural limit hand instead (All Honors or Big Three Dragons). So without breaking either of these rules, the most faan possible to earn from honor tiles are having a pung of your seat wind (when that is also the round wind), two pungs of dragons and a pair of dragons.

Pung of Dragons + Pung of Dragons + Little Three Dragons + Pung of Seat Wind + Pung of Round Wind = 6 faan

FAAN FROM BONUS TILES

Since having all eight of the bonus tiles is a natural limit hand, collecting seven of them instead would grant you the most faan. In particular, if you collect both of your own bonus tiles and any five others, you'll get the maximum amount of faan from bonus tiles.

Own Bonus Tile + Own Bonus Tile + All Flowers/Seasons + Seven Bonus Tiles = 6 faan

FAAN FROM COMPOSITION

Even though a Full Flush is worth 3 faan more than a Half Flush, we'd lose the extra 6 faan from honor tiles if we aimed for one of those hands. So instead, aiming for an All Pungs and Half Flush hand gets us the most total faan here. While Seven Pairs would get us the same amount of faan as Half Flush, we'd also lose all our honor tile faan for that hand.

All Pungs + Half Flush = 6 faan

FAAN FROM WINNING

Many of the yaku in this category are mutually exclusive, so there's a lot to look at. Since our hand is All Pungs already, we cannot have both a Concealed Hand and Self-Pick (since this would be the Four Concealed Pungs limit hand). If we end up making two of our pungs into gongs, we can score a Gong-On-Gong and Self-Pick hand (a replacement tile based yaku can never be combined with Last Tile Win, so this is the maximum we can get from this category).

 Self-Pick + Gong-On-Gong = 3 faan

Therefore, the absolute upper-limit for a single hand is to score 21 faan.

Now, if we wanted to see how many points we could score with one hand, we would instead take a look at natural limit hands, and see how many of them can be combined at once. This is fortunately rather easy, since many of these hands are mutually exclusive. We simply need to make a concealed gong of all four winds with a pair of dragon tiles during the eighth dealer keep (Making a concealed gong voids the possibility of earning a Gift of Heaven or Gift of Earth, so you cannot score that and a Four Gongs limit hand at once. Additionally, since you are obligated to declare a win as soon as you draw the final bonus tile, you cannot score that limit hand with most of these either. Of course, you could replace the Four Gongs with a Gift of Heaven to the same effect).

Four Concealed Pungs + Four Gongs + Big Four Winds + All Honors + Eight Dealer Keeps = Quintuple Limit Hand

If you self-pick your winning tile, you'll earn 1.5 times the points compared to winning on a discard, so let's assume that that is the case. A single self-picked limit hand earns 128 points from each other player, so this quintuple limit hand earns you five times that: 640 points from all three other players, for a total gain of +1920 points in one hand.

Good luck scoring that one.