Can I test my oven temperature without an oven thermometer?

First, buy a thermometer, they're cheap. This one topped America's Test Kitchen testing: Thermometer

Barring taking the plunge and spending $6 on an oven thermometer, this method comes straight from an online class I'm taking from Harvard through edX.org. Course Description

To test the oven, you can either use an oven thermometer, or you can find a material that has a transition at some critical temperature, and then tune your oven near the critical temperature where the transition occurs -- to find out if it actually happens.

Jeff Potters has suggested a wonderful idea in his excellent book Cooking for Geeks. Namely, he suggests that oven testing can be done with melting sugar. His protocol is here.

The calibration is based on the idea that the melting point of table sugar (sucrose) is 366 F (186˚C), [google "melting point sucrose"] and so if you put the temperature just above 366, the sugar should melt. And if you put it just below 366 F it should not melt. It is important when doing this procedure to not use a lot of sugar, as otherwise the heating time for the sugar could be too long.

Materials:

~ 1/2 tablespoon granulated sugar

Procedure:

1) Preheat your oven to 350F (177˚C).

(2) Place the granulated sugar in an oven proof dish or on aluminum foil

(3) Place in oven for 15 minutes

(4) If the sugar doesn't melt, raise the temperature of the oven by 10F (5˚C) and repeat the experiment.

(5) Continue raising the temperature until you find the temperature setting where your sugar melts. Let's call this temperature Tmelt.

(6) You have thus found that the sugar melts somewhere in the interval Tmelt-10, Tmelt. (i.e., if you found that the sugar melted at 400F, but not 390F, the melting temperature is between 390 and 400.)

(7) Your measurement suggests that your oven has the approximate calibration curve

T = Tdial + 366 - Tmelt (if you use degrees Fahrenheit)

T = Tdial+ 186 - Tmelt (if you use degrees Celsius)

where Tdial is the temperature of the dial. Note that if your dial says that sugar melts at 400, this formula states that the actual temperature in the oven is the temperature that the sugar melts (366F)."

Here is another method, this one is from Cook's Illustrated, AKA America's Test Kitchen. I copied and pasted the technique from Water Temp Test

Preheat oven to 350F. Start with 1 cup of room temperature water and check its temperature with an immersion thermometer. Adjust the temperature with either warm or cold water to make it 70F. place in oven for 15 minutes. Check temperature again and it should read 150F. Adjust oven temperature accordingly.

Another way to check oven temp is to simply boil water. Put a couple of cups of water in an oven proof pan and set the oven temp to 200F or 90C, the water shouldn't boil no matter how long you leave it in the oven. Set the temp to 225F or 110C. Eventually, the water should boil.

All that said, buy a thermometer.