You should look at the details of the logic data sheet. The maximum current is frequently quoted for the highest voltage that will be interpreted as a logic zero. There is probably also another absolute maximum current limit.
You can also parallel two HC gates in the same package to double the output current ratings.
Most of logic ICs are designed in such a way that you can safely shorten their outputs to GND line (but not to +V) so don't worry you wll not damage your driver and, as Flatulent said, you can parallel 2 or more HC gates if you wish to drive 1, 2 or more high current LEDs (each with R).