I believe it's a lightning ground. There are some good threads in the past here, do a search and you will probably find some other pictures and info too.
Do what you want, but I personally would not touch one of those things. There is an array I can see up the road, in a big ol field... not worth it to me though.
I would be more concerned about the AM and guyed (if it's short) features. I know someone who jumped the 'Blowtorch of the West' who claims he hasn't felt/thought the same since...
They are not lightning transformers they are lighting transformers. It's an isolation device to protect the circuit providing power to the lighting system on the antenna and the circuit providing RF power to the antenna from each other.
They can definitely arc. That's why they have the two small orbs off to the side with an air gap between them, so that any arcing that happens is between those, bypassing the windings of the transformer.
The arc doesn't necessarily have to be due to lightning either. If the atmospheric conditions are just right (or wrong depending on how you look at it) you can get arcs from the RF power going into the antenna.
AM towers are mast radiators. That is, the tower itself is the antenna. The tower therefore has to be isolated from ground. If the tower were grounded, the signal being transmitted through it would have a short circuit to ground. This is why you can't just climb onto an AM tower while it's live. If you touch the tower while still standing on the ground, your body becomes a conductor to ground. I wouldn't recommend jumping an AM antenna.