The datasheet is a little confusing. If I'm reading this right, on page 1, C1 isn't necessarily required. Then, Figure 11 on page 10 shows the typical application for a fixed output regulator (which this is), and there are no capacitors. Perhaps they were just omitting them under the assumption that it was understood that they are needed? Anyone know?
http://www.ti.com/lit/ds/symlink/lm1085.pdf

 

The answer is YES.

You can try it without one. If it oscillates then you'll know why.
Omit it at your own risk.

 

7.4.2 Stability, output cap, YES
Page one, drawing, input cap, YES if the chip is more than a few inches away from the big DC filtering capacitor in the power supply.

 

So... I'm using a 3.7V battery to power the LM1085 (thus, no DC power supply with a filter cap). The LM1085 is in turn driving a simple op amp circuit and powering a motor. I figure that doesn't change anything and the answer is still yes - add two 10uF caps?

 

I'm using a 3.7V battery to power the LM1085

Page 7, Fig. 1
There is no condition where the chip uses 0.4 volts and still functions.

 

F7U12...

So, what SHOULD I be using to step a lithium battery down to 3-3.3V with a max current of 1.5A (maximum stall current + a little extra for my PWM to be safe)? I looked at step down converters, but a linear regulators seemed easier.

 

Read the specs on your circuits. Maybe 3.7V is fine.

 

I hooked it up to test it to see if maybe I would get lucky, but to get even 3V, I need 3.9V. It plummets below that. Good thought, but no dice...

 

MrChips means, read the specs on whatever it is you're trying to power with a battery. Do the other parts say, "Requires 3.300 volts. Bursts into flame at 3.301 volts."