Inspired by a comment from this question:

What do we consider "uninformative" in a prior - and what information is still contained in a supposedly uninformative prior?

I generally see the prior in an analysis where it's either a frequentist-type analysis trying to borrow some nice parts from Bayesian analysis (be it some easier interpretation all the way to 'its the hot thing to do'), the specified prior is a uniform distribution across the bounds of the effect measure, centered on 0. But even that asserts a shape to the prior - it just happens to be flat.

Is there a better uninformative prior to use?

[Warning: as a card-carrying member of the Objective Bayes Section of ISBA, my views are not exactly representative of all Bayesian statisticians!, quite the opposite...]
In summary, there is no such thing as a prior with "truly no information".
Indeed, the concept of "uninformative" prior is sadly a misnomer. Any prior distribution contains some specification that is akin to some amount of information. Even (or especially) the uniform prior. For one thing, the uniform prior is only flat for one given parameterisation of the problem. If one changes to another parameterisation (even a bounded one), the Jacobian change of variable comes into the picture and the density and therefore the prior is no longer flat.
As pointed out by Elvis, maximum entropy is one approach advocated to select so-called "uninformative" priors. It however requires (a) some degree of information on some moments $h(\theta)$ of the prior distribution $\pi(\cdot)$ to specify the constraints$$\int_{\Theta} h(\theta)\,\text{d}\pi(\theta) = \mathfrak{h}_0$$ that lead to the MaxEnt prior
$$\pi^*(\theta)\propto \exp\{ \lambda^\text{T}h(\theta) \}$$
and (b) the preliminary choice of a reference measure $\text{d}\mu(\theta)$ [in continuous settings], a choice that brings the debate back to its initial stage! (In addition, the parametrisation of the constraints (i.e., the choice of $h$) impacts the shape of the resulting MaxEnt prior.)
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 by maximising the Kullback distance between prior and posterior. In the simplest cases with no nuisance parameters, the solution is Jeffreys' prior. In more complex problems, (a) a choice of the parameters of interest (or even a ranking of their order of interest) must be made; (b) the computation of the prior is fairly involved and requires a sequence of embedded compact sets to avoid improperness issues. (See e.g. The Bayesian Choice for details.)
In an interesting twist, some researchers outside the Bayesian perspective have been developing procedures called confidence distributions that are probability distributions on the parameter space, constructed by inversion from frequency-based procedures without an explicit prior structure or even a dominating measure on this parameter space. They argue that this absence of well-defined prior is a plus, although the result definitely depends on the choice of the initialising frequency-based procedure
In short, there is no "best" (or even "better") choice for "the" "uninformative" prior. And I consider this is how things should 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. (At least before observing the data: once it is observed, comparison of priors becomes model choice.) The conclusion of José Bernardo, Jim Berger, Dongchu Sun, and many other "objective" Bayesians is that there are roughly equivalent reference priors one can use when being unsure about one's prior information or seeking a benchmark Bayesian inference, some of those priors being partly supported by information theory arguments, others by non-Bayesian frequentist properties (like matching priors), and all resulting in rather similar inferences.
