How many ways are there to defeat bowser in Super Mario Bros
The Super Mario Bros manual said of Bowser
"There are several ways to kill him but you only get points if you use fireballs"
To me the word "several" implies more than two but I'm only aware of two ways of killing him.
- Hit him with fireballs.
- Get past him and hit the switch that opens the bridge.
Are there any other ways?
No.
Basically there are limited things you can do to enemies in the game; restricted to jumping on them, using fireballs, or colliding with them while using invincibility. Now since there is no invincibility near the Bowser fight, and colliding with him (whether you jump on him or not) causes player damage, you're limited to Fireballs, and hitting the lever.
The only argument against this however is in the case of the "Imposters" or "Fake Bowsers", which will revert to their original forms once being hit by a fireball and can be dealt with in their own respective manner (e.g. the Goomba can be squashed).
EDIT: @Michaellogg has pointed out that this last statement is actually incorrect; the "transformed sprite" is purely aesthetic, as hitting it with fireballs kills it outright.