Bindweed I have found likes a gentle tug (no EA intended) and it often gives up a decent length of root. I am now getting to grips with it (again no EA) and slowly it is being removed by that method rather than by digging.
Brambles, I am digging out.
Buddliea went a few years ago when I discovered its appetite for growing.
Ivy is a constant fight but I am slowly getting it out of the ground and off of my stone wall by pulling it slowly and pulling lengths of roots out.
Strawberry plants planted by my sister about 8 years ago are the new bindweed. And before anyone says leave them and eat the crop, in 8 years we have eaten 4 or 5 small strawberries. Pah!
Passion flower is another pain. I have one area left where I need it destroyed as it keeps throwing shoots up. I love the flowers but it is more intrusive than a fairly intrusive thing.
Didn't the US of A use a deforestation liquid in 'nam? I am reluctant to napalm it as I will harm the nearby lavender.
The major battle is that tall plant with nice looking purple spiky flowers (Willowherb IIRC). It's the Devils own job to get rid of that. I usually pull the group of stems at ground level, but at times its resistance is very good.
Slowly the garden is being cleared of these weeds, then as areas of soil are revealed the cats come around and make their deposits, despite the best attentions of my cat hating dog.
Cleavers or Sticky Willie (calm down Mick) is another pain this year. Creeping in from our neighbours.
And finally Esther, Elder or ground elder, that is pretty hard to eradicate.