Anything that doesn't look like a leaf or a leaf bud is probably a flower, the start of flowers, or the resultant seed pods.

FataMorgana

Your basil is putting up bloom shoots. If you want to extend the harvest life, try cutting back hard and then give it some liquid fertilizer every couple of weeks. Especially if it is grown in a container. I do this multiple times per season, and the plants produce another batch of the tender and tasty deep green leaves.

Thanks. The weather's been so hot these things seem to bolt non-stop...

I do hope you are going to let at least one plant to go to seed as fresh seeds will provide you a lot more plants next year. Much better than any store bought seeds. They do not even need to be harvested over the winter if you just do not have time to harvest them once the dry out.