In 4-Manifold theory one makes often the use of Kirby Diagrams to construct 4-manifolds (compact or non-compact) with specific gauge and topological properties (for example small betti numbers, spin structure, etc.).
This raises a couple of questiona:
1.Is any compact or non-compact 4-manifold obtainable as a (finite or infinite) handle diagram ?
2.What are the properties needed for a compact or non-compact 4-manifold to be represented as a handle diagram ?
3.What are examples of 4-manifolds with no handle diagram ?
The diagrams can be as complicated as you want (so 0-, 2-, 3-, 4-) handles can be present. I do not know if you can get rid of all the 3-handles in the non-compact case.
This question came forth from the discussion explicit "exotic" charts . I am trying to get help of more people on that, then putting those things in comments (the question of explicit charts of an $\mathbb{E}\mathbb{R}^4$ is another one, albeit interesting in it's own right).
The question is answered by Bob Gompf by email, see my comment for the main part of his answer.