Based on the research I've done online, I believe this unwelcome plant to be Nodding Spurge (Chamaesyce nutans, syn. E. nutans). Nodding Spurge begins life as a tiny pair of leaves about the size of this comma , and that may be the only time in its growth cycle that you can scrape it out or hoe it up without fear of reprisal. Once it begins to develop those tiny pink-tinged white blooms, you'll want to pull or dig these weeds out.
Nodding Spurge can grow up to 3 feet tall; the long thin stems enable it to grow up through other plants and hide therein ... by the time you spot it, who knows how many seeds have already fallen? Take care when pulling or digging it up to avoid releasing more of the tiny seeds.
According to an article on a Virginia Tech website, Nodding Spurge is very similar to Hyssop Spurge (Chamaesyce hyssopifolia). Looking at the pictures on that page, I believe that I may also have Hyssop Spurge, mainly in and near the crushed granite paths. I had noted a difference in stem and leaf color in seedlings in those areas, which I attributed to soil conditions. I'm not willing to nurse a seedling along in the interest of science, however!