The "uninformative" prior is sadly a misnomer. Any prior distribution contains some specification that is akin to some amount of information. Even the uniform prior. Indeed, the uniform prior is only flat for one parameterisation of the problem. If one changes to another parameterisation (even a bounded one), the Jacobian comes into play and the prior is flat no longer.
As pointed out by Elvis, maximum entropy is one approach to select so-called "uninformative" priors. It however requires (a) some level of information on some moments of the prior to specify the condtraints and (b) the choice of a reference measure in continuous settings, which brings the debate back to its initial stage!
José Bernardo has produced an original theory of reference priors where he chooses the prior in order to maximise the information brought by the data. In the simplest cases, the solution is Jeffreys' prior. In more complex problems, (a) a choice of the parameters of interest must be made; (b) the computation of the prior is fairly involved. (See e.g. my book The Bayesian Choice for details.)
In short, there is no "best" (or even "better") choice for "the" "uninformative" prior. And I think there should not be because the very nature of Bayesian analysis implies that the choice of the prior distribution matters. And that there is no comparison of priors: one cannot be "better" than another. The conclusion of José Bernardo, Jim Berger and many other "objective" Bayesians is that there are roughly equivalent reference priors one can use when unsure about one's prior information, some being backed up by information theory arguments, others by non-Bayesian frequentist properties, and conducting to rather similar inferences.