Darlington Transistor Array
Sure, just copy the circuit inside the ULN2803 using (lots of) discrete parts. 



For push-pull drive from the micro and LED drive you can probably leave out the diode (and the parasitic diodes with the dashed lines, of course), as well as the 3K and 7.2K resistors, so just the two transistors and the base resistor. 

The maximum current will be set by the transistor Ic(max), for the 2N3904 that's only 200mA. The ULN2803 is good for 500mA, but whatever your requirements are.. you could use a 2N4401 and either another 2N4401 or a 2N4401 and a 2N3904. 

Edit: Given the schematic in the link, you don't need to duplicate the current capability of the ULN2803. A single transistor with a 510 ohm base resistor (assuming 5V drive) should suffice, so 8 transistors and 8 resistors. The maximum collector current is about 120mA (at 12.5% duty cycle) so a 2N3904 or 2N4401 per channel will do.