The exact spot where the Missouri River empties into the Mississippi is located inside Columbia Bottom Conservation Area, a 4,256-acre property north of St. Louis managed by the Missouri Department of Conservation. There's a viewing deck at the confluence. I stood on it for a long time and watched two enormous rivers become one, which sounds like something you'd see and shrug at, but doesn't actually work that way in person.
The Missouri comes in carrying silt and force, visibly different in color from the Mississippi. For a stretch downstream you can see the two rivers running side by side within the same channel, not quite mixed, the line between them blurring slowly. North America's two largest rivers. The point where they meet feels like it should be more heavily marked.
Columbia Bottom is primarily a birding destination — it's on the Mississippi Flyway and has recorded 285 species on eBird, which puts it among Missouri's best birding sites. The managed wetlands attract herons, egrets, and pelicans during migration, and the fields and hedgerows hold warbler and sparrow diversity in spring and fall. I walked several miles of the road network that crosses the area, stopping at each wetland pool to see what was using it.
In August the light at the confluence goes flat and hazy, not ideal for photography but fine for just being somewhere enormous and real. Go at dawn in spring if birds are the goal. Go any time if the confluence itself is what you're after.