If you have an alternator that is hooked up, then you have to have a battery. A battery acts as a sort of voltage stabilizer in the electrical system. The voltage regulator controls how much the alternator puts out based on battery voltage. If you take the battery out of the equation, the charging system will think that the battery is totally dead, and will be charging at maximum capacity all the time. Otherwise known ass full fielded.
Not only will this quickly ruin your alternator, but a lot of other things in your electrical system that are not designed to be run at the higher voltages your system will be operating at. Keep your battery hooked up.
How did you plan on operating you starter anyway?