There exists a jupyter kernel for cling, which allows this (also this is C++ not C)

Yes, and it is quite good: https://github.com/root-project/cling#jupyter

I find it invaluable

Here is another one in development: https://github.com/QuantStack/xeus-cling

It would be interesting exercise to compile cells into object files as they are entered and load them dynamically into a persistent process, doing needed linking etc on the fly in-memory. You'd also need to generate a header file from the current in-memory symbol table and implicitly include that in compilation.

That might actually work relatively well for simple cases. But I imagine there would be a ton of problematic edge cases, considering that with C you can end up with all sorts of weird stuff.

The %cython mode for Sage notebooks (which I wrote in I think 2008??) does precisely that by generating a random-named Python shared object libary and importing it into the runnig Python process after each evaluation. There's a similar (%%cython?) mode for Jupyter notebooks.

If you protect it appropriately, you've basically re-implemented exec.

However, every cell would need a main() function or it wouldn't do anything. Or you hold onto a main function from previous cells, perhaps overriding it, and make the notebook confusing as hell, defeating the point of notebooks

It does not yet :)