garden compost bin

Depends on how hot the contents of your bin get. The conical black bins are great when in the sun as the contents are 'cooked'. You are taking a big risk of spreading weeds if you put any of the following in a bin, I'd say:
1. Horsetail /marestail. ..equisetum. These are a 'prehistoric' plant and the roots are just about immortal!
2. Bindweed.. those bits of root seem to be resistant to anything, though turning them to sludge in a black bin bag first seems to work.
3. Dock roots as the big ones seem to come back to life in the bin. (do bin bag treatment also)
4. I've been told that poplar leaves restrict growth in other plants, but not found this if they are well rotted down.
If you've the patience, why not put all the 'nasties' together in a black dustbin and add water. This would make a great foul smelling feed which would be good for tomatoes and things, and when fully rotted, the weeds will be harmless when added to your heap.
Just about anything else can be composted as long as you turn it and get some high temperatures. I compost everything (ignoring my own advice above!) but I do get some weeds coming through, but I find they aren't too hard to pull out when I spread the compost if any have started into growth.
I'd definitely not compost anything with pernicious fungi like white rot on onions though as you will spread it to other parts of the garden

I'd agree with the above, but would add that, if you're not "hot" composting (you're just chucking it in the heap and letting it get on with it, not bothering to turn, etc.) then definitely don't put weed with seeds on, definitely not pernicious weeds like bindweed, etc., as mentioned above, and not woody material - it'll take years to rot on a cold compost heap. I was also advised that fruit (whole apples, oranges, etc,not the peelings) should be composted separately, as should leaves (from autumn leaf fall) if you have a lot - these go in black bags, wet, tie the top, couple of holes in the bottom, leave to rest somewhere quiet for a couple of years.

Deffo not your cat...mine crawled in for a snooze after I'd left the lid up to dry it out a little & I threw in the sloppy contents of the kitchen compost bin...she was not amused.

And can I just say if you are turning the compost heap please be careful. Today I evicted a little family of mice from mine. Hopefully, following Bonkersbon's advice, they will be ok.