I think here it's mostly pigweed. When I moved in the entire garden was overgrown with it, waist high. And I'm assuming that if I work assiduously at weeding and mulching I will have it down to a manageable nuisance in about, say, a decade.
There are a number of native grasses that I tolerate in one place but that pop up in others.
Then there is white yarrow. The first year I cultivated it. The second year I tried to confine it to its own space. Next year I think I will be treating it as a weed -- except in one or two specific places where it seems to be the only thing that will grow except for pigweed. I wouldn't be surprised if in three more years I will add it to the eradication list.
FInally there is sweet clover. It will, in eight or ten weeks grow chest high and two feet across if it gets any supplemental water. While the garden is young and full of weeds I tolerate it because I happen to be gaga about the smell of it, but I do try to dig it up before it sets seed. I enjoy adding its dried leaves to the holes of my new roses.