Perhaps 1/(x^4+x^2) is the simplest (in some sense) example satisfying @Erling’s challenge criterion for which a suitable conic formulation has not yet been identified? That should put it at the top of the leaderboard.
Unlike Express 1/(a^2+x^2) in cvx 1/(x^4+x^2) is convex over its entire domain, so does not get demerit points for not being convex over its (natural) domain. Its only “non-niceness” is being singular in the middle of its domain, at x = 0.