What was Gödel's Constitutional Problem? It is well known that Kurt Gödel had doubts concerning the US constitution and believed that it somehow was inconsistent and opened up for a dictatorial grab. 
What was he thinking?
 A: The history of how Gödel studied the US constitution in preparation for the citizenship exam is well documented by Jeffrey Kegler, as pointed out in the comments. Perhaps more relevant for MO (as opposed to HSM) is what the flaw in mathematical logic might have been that Gödel had noticed in the constitution.
A scholarly account of the latter issue is given in Gödel's loophole by Enrique Guerra-Pujol  (2012).

The mathematician and philosopher Kurt Gödel reportedly discovered a
  deep logical contradiction in the US Constitution. What was it? Although no one knows with certainty of what Gödel’s loophole or “constitutional contradiction” consists, in view of Gödel's research interests it is reasonable to search for a problem of self-reference. We will refer to a constitutional design defect as “Gödelian” if the flaw is contained in a constitutional statement that is self-referential.

.

We have
  identified one particular “design defect” in the Constitution that
  qualifies as a “Gödelian” design defect. In summary, Gödel’s loophole
  is that the amendment procedures set forth in Article V self-apply to
  the constitutional statements in Article V themselves, including the
  entrenchment clauses in Article V. Furthermore, not only may Article V
  itself be amended, but it may also be amended in a downward direction
  (i.e., through an “anti-entrenchment” amendment making it easier to
  amend the Constitution). Lastly, the Gödelian problem of
  self-amendment or anti-entrenchment is unsolvable.

