It looks like the (ULN2803A) package can dissipate a maximum of about 1.8 watts total at 25C ambient (room temperature). You might be able to increase that a bit by using heavy traces on the PCB, or as heavy copper wire as you can if you are hand wiring. But if you push it to it's maximum, it will be too hot to touch.
Just a suggestion here, but I would keep the total package dissipation to below 1 watt. The spec sheet gives the Vce as around a volt if you keep the current on any given pin below 300ma and that would suggest a maximum, if load is spread out across the device, of about 1 amp total. If you limit your LED drive current to 10ma, that would suggest you could run 100 LEDs per device with no more than about 30 on a single output. For 20ma, cut those numbers in half.
As has been said , if you needed more LEDs than that you could stack them in series and use a higher drive voltage. As an example, if you re using white LEDs with a forward voltage of 3.5V and drive them with 12V you could use 3 in series with a 150 0hm resistor to drive them all at 10ma. But since you said you won't need more than 15 per output, that shouldn't be a concern.