I've read tons of posts on similar issues but none match my scenario exactly so I'm hoping for some advice.
I have an 05 TL with 180k miles that I've owned since 120k.

I replaced my battery a few weeks ago. I've had no problems until 3 days ago. I drove to a store, was there for half an hour, came out and my car wouldn't start. Turned the key, lights radio etc turn on, but I turn the key and hear nothing. Headlights and dash didn't change but the radio switches off until I let off the key. I assumed it was loose battery cables since I had dealt with that before a few years ago.
I re-attached the cables, jumped it off, nothing changed. I fiddled with a few things and tried again, no luck. Chatted with a stranger, got back in the car without really touching anything, and it started right up like normal.

I assumed it was the starter and took it straight to Advance Auto to have them do a diagnostic test. Had to turn it off multiple times during the test and it cranked up perfectly each time. Battery, starter (surprisingly!), etc all checked out perfect, and the guy suggested it was possibly corrosion on the inside of my battery cables, or a bad ground / starter connection.

Today, I had the same issue leaving work. After 2 failed attempts, I got out, wiggled the battery cables around, got back in and it cranked perfectly. Probably just a coincidence?

When I got home, I grabbed some of the corrosion spray, battery cable / post cleaning brush. I removed the cables, sprayed them a bit, brushed them and the posts, and re-attached. Since then, I've tried many times and have yet to get the car to start.

Things I've checked:
I hear a quiet click when trying to start the car down near the fuse/relay box below the steering column, which I think means the starter relay in there is working properly?
I've tried all of the suggested shifting to neutral/park/etc strategies I've read, no change.
I did a quick spot check of the condition of the ground cable and connection to frame, as well as the positive cable and connection to fuse box and starter. All appear fine, but it was dark by the time I was looking so I'll check again tomorrow.

Any tips? Is it possible that the starter diagnostic test was just wrong?

Thanks!

 

When it dosent start have someone hold the key while you tap on the starter,if it cranks replace it.

 

Mine was the ignition switch

Ignition Switch

 

Well, did you confirm that your battery and ground cables are not corroded on the inside?

 

Have you tried replacing the + battery cable? At this age, it may be corroded internally where you can't see not allowing enough starting juice through. I had the exact same starting problem a couple of years ago and replacing the cable did the trick.
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As DMZ mentioned it is crazy how many issues can be caused by corroded positive cable.

If you want to check things out here is a starting diagram.




Anything going bad here and car won't start. There is couple possibilities. If you have a test light and car will refuse to start you can check what is the problem.




To eliminate the starter cut relay, you might switch it with power window relay. They should be the same type, and power window relay is not that important. If windows will randomly stop working then you know what was bad.
Starting relay is #18, power window relay is #22.

If car will still refuse to start I would check if started solenoid is getting good power. If starter solenoid has good power but doesn't crank, and starter main power is also good, then starter is bad.
If starter solenoid doesn't get power, probe relay #18 pins to check if it gets power. If relay doesn't get power check ignition switch.

 

OP has not posted again so I doubt we'll ever find out how it turned out.