Of course, this can't happen - because the largest one we know about was only 40cm long. And they've been extinct for almost 200 million years.
The fossils known as conodonts are amazingly common. A bottle of vinegar and a handful of limestone rubble from any rock of the right age is almost bound to include them, and if you live in a good area you can start a collection from your nearest road cutting. Like the forams I talked about earlier, they're vitally important index fossils, with the added advantage that they change colour depending how hot the rocks got in their history. They swarmed in all the world's oceans for some 300 million years.
But until 1983 science had no idea what they were actually from.
Various candidates were proposed, including annelid worms, arthropods, molluscs, and even plants, although it was sometimes suggested that they were fish teeth. Certainly, they were made of calcium phosphate - the same material bone and teeth are built from ( Old conodonts never die. They just lose their apatite ). Odontogriphus from the Burgess Shales was pushed forward as another possibility, as were arrow worms. The latter are planktonic carnivores that cruise around like guided torpedos of toothy death. Indeed, it now seems that the Protoconodonta really were related to arrow worms - but the rest of them are something else entirely.
The 1983 fossil from Scotland was quite a surprise - because the only hard part of the conodont animal were the aforementioned conodonts, the rest of the body was only preserved under unusual circumstances. And even then, not well. Nonetheless, the fossil had the chevron striations of chordate muscle, traces of a notochord, and a cute little tailfin with rays! All pretty much conclusive that the thing was one of our distant relatives, if not an actual vertebrate! ( Which makes the absence of an obvious skull slightly puzzling )
Scanning Electron Micrographs of some fairly broken conodont elements -
Much better pictures here
Model of the jaw assemblage of an ozarkodinid conodont, here
For more on conodonts, see "The Jaws That Catch": an Introduction to the Conodonta - it gives some good reasons why the two things at the front are not necessarily eyes, and has an admirable amount of snark as well
As an aside, conodont biology has been the source of one of the strangest social transformations in recent academic history. Consider that, for many years, conodont affinities were unknown, even as to Kingdom, much less phylum. However, conodont elements are very common items and easy to isolate. As a result, conodont elements became key stratigraphic markers. The students of these obscure phosphatic scraps were thus much in demand as petroleum stratigraphers. While stratigraphy is a relatively low academic calling, it pays rather well because of its obvious real-world commercial value. By contrast, vertebrate paleontology has always been a useless, but popular and respected pursuit, although the pay ranges from the execrable to the non-existent. Imagine, then, the dismay of all concerned when conodonts were discovered to be vertebrates. Suddenly, conodont workers were dragged from the oil-stained proletarian legions of Houston and Riyadh petro-imperialism and thrust into the rarified, if ragged, intellectual circles of vertebrate palaeontologists.