There are two questions here: first, can Descartes' cogito be doubted—is it open to doubt that "I" exist? Second, more generally, is there anything that's not open to legitimate or reasonable or rational doubt? (What people are psychologically capable of doubting maybe another matter.)
On the first, may philosophers would say yes. Even if it's certain that there's thinking going on, it doesn't follow that there's some one or some thing doing the thinking. Consider the Buddhist/ Humean "no self" view. On that way of understanding things, there's no substantial self. There is, as the Humean might say, just a bundle of perceptions. "I think," on this account, is just a manner of speaking. We can't get from "there is thinking" to "I exist."
So maybe it's open to doubt that I exist. Is it open to doubt that there's thinking going on when it seems that there is? Maybe not, though I don't doubt that some clever philosopher could offer an interesting argument to the contrary. What else? Can it be rationally doubted that anything that exists is self-identical? Some might say it depends on whether there are "things" to be self-identical.
At this point, however, I feel more like a philistine than a philosopher; I'm not sure I care. Maybe there's something beyond all possibility of rational doubt, or maybe there's not. But even if there isn't, real epistemic life will go on as it always has. We'll continue to believe things, reasonably in my view, and, I'm willing to say, we'll even know some of them. Here my sympathies are broadly with G. E. Moore. The arguments that one has to come up with to adopt wholesale skepticism are a lot more fragile and open to doubt than the innocuous thought that we actually know some stuff, even if we're not entirely sure just what.