I have installed my 6th resistor and they continue to burn up, leaving #5 only.
You need to replace the blower motor. It is apparently drawing too many amps & continually burning up the resistor.
I agree with cigroller. I’d also suggest that if you’re using an aftermarket part, pay the extra expense and get a dealer part for the next resistor to see if that helps.
I also agree. It’s time to replace the blower motor.
I’ll give another vote for replacing the blower motor. Back in the days when I used to repair my own radios and television sets, the wattage of a resistor would be specified as well as the resistance. There was a reason for not replacing a 1/2 watt resistor with a 1 watt resistor: the resistor served as a fuse. A half watt resistor would burn out to prevent further damage while the 1 watt or bigger might allow too much current to flow through another component.
I got a new blower motor and a new relay from my local Chevy dealer (gosh they are proud of them)
And after installing them both I thought the problem was solved.
But, I was in the habit of turning the blower motor off before shutting down the engine and leaving the vehicle.
But the first time I failed to turn the blower off and left it in the on mode and shut the truck down, the next time I got in it the blower would only run on #5 high. Relay fried again. I replaced the relay again. All was good until I loaned my truck to my brother and failed to tell him to turn the blower off before shutting the truck down and when I got it back I found the blower motor control knob in #3 and it was fried again and would only run on #5.
This has been going on for 7 years? You are a very patient person 
. 7 years ago In your OP you said it was the blower motor resistor that was getting fried, and above you are saying it is a relay. I presume you are still referring to the resistor. If you’ve replaced the resistor and blower motor and it still keeps happening, and you are mounting the resistor properly so the heat sink function of the mounting method is working, most likely either the fan cage isn’t rotating freely, which is loading down the motor, or there’s a problem with the speed selector switch. You might try replacing that.
I should say at this point the “low” heater blower speed on my Corolla hasn’t worked for 20 years. I just use the other speeds that work instead. … lol . .