If you mean how many LED's can be controlled in unison by a single PICAXE output pin with no buffering and only passive components, you are constrained by the PICAXE the current which can be drawn from the pin ( 20-25mA ).
As Michael 2727 says, 5 LED's in parallel drawing 4mA each will work. You can also connect them in series as well as in parallel then you are limited by the voltage they are driven from, the PICAXE supply voltage if connected between the PICAXE and 0V but you can have more if the LED's connect between a higher voltage and the PICAXE.
You could probably drive 15 LED's from a PICAXE running in series and parallel with a 5V supply, but there's a maximum current for all eight outputs combined so you could not do that for all pins.
The number becomes almost unlimited if you use transistors or FET's controlled by the PICAXE and have a suitable power supply.