Is it possible to connect 9 blue leds to a 9 volt battery?
Of course it is possible. Whether you want to do that is another thing.

9V batteries can have more than 9.5V when new and are considered empty around 5V.

Blue LEDs have approximately 3V forward voltage, but it won't be exactly 3.0V and it will vary due to manufacturing tolerances etc.

So assuming all the LEDs have exactly 3.0V forward voltage at their rated current, a fresh battery with 9.5V connected to 3 LEDs can give the LEDs quite high current so that their useful life is reduced, or they might burn out immediately.

And assuming the battery voltage has dropped to say 7V, maybe no LED will glow even if there is still energy left in the battery.

Also the LED manufacturing tolerances will cause the LED chain with smallest sum of LED forward voltages to be brighter than the other chains, and maybe the other LED chains won't light up at all.

So, in light of this, maybe you don't want to use chains of 3 LEDs without resistors on a 9V battery, even if it is possible.