My 85 Vette will not start.  A few weeks ago I came out to start it and the battery was dead.  We jumped the battery and everything was fine.  I tried to start it yesterday and again the battery was dead.  I tried to jump it, no success.  I called AAA and they came to jump start it.  NO success.  The posts on the battery were corroded so the guy suggested changing them.  I changed the battery posts and AAA came back out.  He tried using his truck and the portable charger and it still wouldn't start.

Is it multiple clicks like a buzzer, or a single click? Do the lights stay on when it 'clicks'? Or does everything go out, then come back on after releasing the key from Start?

If it's a buzzer-like click, it is most likely not the starter. But still a corroded cable connection, or cable.

If a single click, and all the lights stay on, this may be a starter, or the solenoid on the starter.  Starters last a long time. Usually long beyond 100K miles, so if it's less than that, be double-sure it really is the starter itself.

Sometimes corrosion works its way down into the cable below the battery connection. Close inspection may reveal that the insulation is bulged for a few inches below the 'eye' of the connector. Working the cable while 'feeling' the bulge with the other hand, may detect a 'mushiness' of the underlying conductor. I've had several side-terminal battery cables that had this bulge, and when cut open the conductor was nothing but green dust. The bulge and mushy feeling when flexing the cable are the clues.

Because the main cable is to the starter, and the little wires are for everything else, the rest of the car may seem normal, but only the starter doesn't crank. Checking the voltage reading at the big terminal at the starter while in Start position will reveal if it's a voltage supply problem, or the starter itself.

There's some ideas.
Dave

Thank you for the information.  I will go and check it out

Well, I cleaned the corrossion off of the cable connections.  The cables themselves are solid.  After cleaning the cable connections and trying to jump it again, the clicking sound went away.  It just sounded like it didn't want to start this time.

Ten A C,

Thank you.  I will have the mechanic take a look at the wires.

If the Security System is activated, the starter-interrupt relay will cycle when the key is turned to START. This would be a single click when turned to START, and again when released.  In an 85, this relay is located behind the radio.

Does your factory alarm function correctly?

And to use proper semantics, your problem is a "no crank", not a "no start" condition. There's a difference in the diagnostic approach.

I'll bet it is a starter, or cable problem. The anti-theft system would flash the "Security" light if that was causing the no crank issue. Dead battery-jump start-dead battery again...I'd be looking at the wiring around the starter, as has already been suggested. Doesn't take much corrosion, or looseness in the connections to cause this.
Good luck, and let us know what you/they find.

Thanks.  I'm not aware that my car has an alarm system.  I've never seen any lights indicating such.  How would I know if it does have an alarm system?  The previous owner never mentioned it.

Before you do the starter, try locking the doors and unlocking them....my first day I tripped the anti-thieft device and got the same clicking which turned out to be the horn relay...I had the horns disconnected so the horns never went off....might save you a few bucks.....