I remember a long time back stumbling across Akodo's Technique, referring back to the old Avalanche of Blades Meditant, and thinking, "Hey, this would let you make a lot more attacks." And it did. It added several attacks to the chain before Avalanche would again miss and break the chain.
So now, enter the Kata, The Victory of the River (The Way of the Samurai, in the Rokugan series; pages 12-13).
You need Cleave, Great Cleave, Power Attack, and to burn 2,250 XP. Thankfully - and charmingly - XP is a river, after all.
You also need to be using a katana or a daisho. I realize that part is a bit of a bummer (yes, yes, taking Cleave and Great Cleave is also a bummer).
However, I promise that it's well worth it (until your DM unleashes his own infinite avalanche of rocks falling and everybody dying). In return, you get, for the 180-minute duration of the kata (which takes 30 minutes to prepare each time, but the XP cost is a one-time expense to learn the kata), a cumulative +5 insight bonus to all additional melee attacks in a round any time you hit an opponent with a melee attack.
Since Avalanche of Blades gives you a -4 penalty between attacks and this gives you a +5 bonus between attacks, you actually net a cumulative +1 bonus between attacks. As long as you can prevent automatic misses on a 1, and assuming your to-hit is high enough to land on every other result (2-20), you'll keep swinging until your enemy is a bloody pulp.
Note: you can still use Akodo's Technique. It's just mean and unnecessary because you don't actually need it to generate more attacks anymore. Instead, it just raises your to-hit to obscene levels at every subsequent attack.