Nikon SB600 Flash not keeping up with Nikon D300?

If I continuously shoot pictures with my Nikon D300 (like 2-3 photos per second), some photos have flash and others don't. It seems the flash can't keep up. I have new Duracell batteries in the flash and my camera is fully charged, so I don't understand what's going on. Is there a setting on the flash or the camera I should change?

I don't think there's anything you can do about it - the flash simply has too long cycle times. Keep in mind, the energy of the flash is not drawn from the batteries in the moment when it is triggered, but already before that happens, over a much longer time than the actual flash outburst takes. The energy is then stored in a capacitor. That's that high pitched noise that you may hear when you switch it on and between flashes.
Recharging the capacitor between two flash bursts takes some time, that's the problem you're running into.

The only thing that comes to my mind is to use less flash output, but that may not be an option.

Duracell alkaline batteries are not a good diet for a sb-600, they last no time at all. Decent, high powered rechargeables from 2500 mah upwards perform much better.

Even with a fresh set of one of those your flash won't keep up with you for long. If you demand a lot of light the recycle time will get you, if not sooner or later the temperature cutout will prevent you from blowing up your flash. I've shot sessions where I had to change flashes every 10 minutes or so to give one a chance to cool down.

For the serious flash user nikon now has the sb-900 with the optional sd-9 battery pack. This video gives an idea how that performs

If you work in a controlled enviroment some sort of studio lighting may be the best way to go. Continous light from lamps will also be more comfortable for your model than rapid fire with a speedlight.

Your shooting faster than the recycle rate on the flash… Try some better batteries or shoot slower…

You should go with Sunpak PF30X Electronic Flash

Small and light weight body requires only 2 AA batteries
Guide Number (ISO 100): 100' (30m)
TTL signals are automatically recognized just by mounting and turning on
EV adjustment buttons enable free operations
Auto Focus Assist Light supports focusing in low light situations

You don't understand the nature of flash photography. Flash needs time to recharge. Flash can't go off over and over again in quick succession indefinitely.

Some flashes can keep it up for a few shots, but then they need to recharge (the same way that cameras have a 'maximum burst speed').It helps if you lower the output of the flash or if you use a battery pack instead of batteries. But even then, there will be a limit to the number of flashes per second.

If you read the manual that came with your flash you would know this…