I was just looking back to some notes that I took a few years ago, when I was reading Etnyre's notes on Legendrian Contact Homology in $\mathbb R^3$ and I happened upon the following question that I had left for myself: Is there a non-trivial Legendrian knot in $\mathbb R^3$ (with the standard contact structure) that is not a stabilization of some other Legendrian, where each generator of the DGA has positive grading?
The standard Legendrian unknot has positive grading and there is a straightforward procedure whereby any Legendrian knot can be given positive grading by repeated stabilization, but I am curious about the existence of other examples. The purpose of this was just to find easy-to-compute examples for Legendrian contact homology, as this condition forces each graded component of the DGA to be finite-dimensional. This is another reason why stabilizations are excluded: the homology of a stabilized Legendrian always vanishes.