The best place to find answers to voltage / current / power questions is the datasheet...
http://www.pjrc.com/teensy/datasheets.html
Search for "Electrical Characteristics"
h4t:
I know there is a maximum limit on how much current each pin can output (~25mA-40mA depending on the chip),
Nope...
30.1 Absolute Maximum Ratings*
DC Current per I/O Pin ............................................... 40.0 mA
*NOTICE: Stresses beyond those listed under “Absolute Maximum Ratings” may cause permanent damage to the device. This is a stress rating only and functional operation of the device at these or other conditions beyond those indicated in the operational sections of this specification is not implied. Exposure to absolute maximum rating conditions for extended periods may affect device reliability.
The AVR processors used for Arduinos consistently have an absolutely maximum per pin current of 40 mA. Try to stay close to half that; 20 mA.
but what about all pins at the same time?
For the entire processor...
DC Current VCC and GND Pins................................ 200.0 mA
Basically, I'm looking to drive 39 individual LEDs directly from the Teensy++ board without multiplexing,
No problem. 200 mA - 30 mA for the CPU = 170 mA for your LEDs. 170 mA / 39 LEDs = 4.35 mA per LED.
Be sure to check the datasheet for "per port" limits.
and I'm wondering if I can do so or if I need to get some Darlington arrays.
If you plan to drive the LEDs at more than 4.35 mA then the answer is "yes".
My hope is that if I plug in a 5V @ 1A power supply to the Teensy, I can utilize all of that power to drive the LEDs via the microcontroller at once. What do you think?
In the words of the great Mr. Lizard, "We're gonna need another Timmy Teensy!" I think you would quickly turn your Teensy into slag.