Well, I'm afraid I do not have much experience about the "cheapest" drive.
Personally, I prefer TEAC, Plextor and Pioneer. The TEAC and Plextor are supposed to be the so called "professional devices", thus maybe a Pioneer drive is much easier to find.
Just few weeks ago, I purchased a brand new Pioneer DVD drive, the model is DVD-130D.
It's not a burner, it does read dvd disc and provides the "analog out" signal.
Actually it's probably Pioneer's last DVD drive which provides the analog signal, and it's probably Pioneer's last IDE interface DVD drive.
I got this drive with an acceptable price -- $5.
Regarding the burner, let's just skip the CD burners, they're way too old and I doubt that they're still functional. So let's focus on the DVD burners.
The last model of Pioneer's IDE interface DVD burner is DVR-118. Which I do NOT recommend, because altough there's analog out interface on its case, this signal is NOT provided! How do I know this? Of cource because I've purchased 3 sets of it, two brand new, one 2nd hand. And none of them could play the CD tracks while playing the DOS games, such as the Quake, the Screamer, the Syndicate... Huh, I must be a real smart **s!!
The last DVD burner of Pioneer, which DOES HAVE the analog out function, I think is DVR-117. It's not easy to find a new set today, but if you could consider a nice 2nd hand drive, 117 and older models are well recommended.
All the other SATA burners and regular drives are NOT recommended for the same reason. Of course if don't mind the analog out function, these drives are still solid choise! Plus it's very easy to get a brand new one SATA drive today.