A "volunteer" plant is merely a weed that is interesting.
I had a fig tree volunteer in my back yard. I tended it for several years, but it never had any fruit. I finally had it removed. Then I fought root suckers from it for about two years.
I found a palm seedling (Washingtonia filifera) in one of my flower beds. Although I really don't like palms in my landscape, I was curious as to how it might grow. I dug it up and put it in a flower pot. Several years later, it's still in its pot on my front walkway.
Once, when I was still trying to have a dichondra lawn in back, I had the lawn renovated. The soil amendment was contaminated with cinquefoil (Potentilla neumanniana). I now have cinquefoil in most of by back yard beds. It seems to cover the bare spots where nothing else will grow and looks nice year round. Now I'm trying to get it to grow in the parkway in front, planting cuttings from the back yard.
The number-one weed in my garden are the seedlings from my evergreen ash (Fraxinus uhdei). See my . No, I will NOT let these grow for a while. A four-inch seedling can have a 10-inch tap root. The 10-foot sapling (about 1-1/2 inches in diameter) that I planted 30+ years ago now towers twice the height of my two-story house and has a trunk almost three feet in diameter (more than nine feet in circumference). Some of its surface roots are more than three inches in diameter; some have grown up under my sprinkler lines and cracked them. I love the shade of The Tree; but no, I certainly would not let these weeds (its babies) grow for a while.