Thanks for very good information.
I have been using the battery that came with it, in conjunction with using my car battery to try to jump it. Let me get more detailed in case anything will stand out.
I bought the mower a few weeks back. When I went to check it out, the guy said it needed a new battery because he couldn't get it to start. With it on a charger, he tried to start it a few times and it the starter began to smoke. I bought it anyways figuring I could have the starter rewound and be in action. I figured that the combination of bad gas and the motor being "gummed up" from sitting and the blade belt turning even when disengaged, the starter was having to work too hard. I got it home, changed out the gas, disconnected the mower deck drive belt, and was able to get it running. The second day that I tried to use it, it wouldn't start, and the battery died after trying unsuccesfully to start the mower. I tried jumping it with the car battery, to no avail and the started was smoking bad and stinking. I ordered a new starter after pulling it apart and seeing all of the windings' insulation melted and black.
The new starter motor worked OK, but not spectacular. The thing has never (except one odd occassion) turned over and over and over. It turns over 1 to 1.5 compression cycles, and then I have to re-key the ignition. But it has worked a few times with the new starter.
The second day of running the mower with a new starter (same old battery) I was mowing and turned it off to hook up the trailer. It was off for 5 minutes and when I went to restart it, it wouldn't go. I tried unsuccessfully until the battery was dead, and tried to jump it with the car again. I stopped and put the battery on the Battery Tender and it charged up to 13+V. It still did not start. Once again the starter began to smoke. I pulled it apart and it doesn't look as bad as the first one. But it doesn't look good.
I pulled the battery out and the starter relay/solenoid and the associated wiring. I replaced all of the 12 or 10 gauge wire with new 4 gauge wire with gold terminals. I had noticed that wire getting really hot. I tested the solenoid and it is fine. Continuity when energized, none when not.
Is there a way to test the starter motor windings to determine if I need to replace it again? How can I trouble shoot the battery to determine if it is truly "hosed"?