Uniform Random on $(-\infty,\infty)$ Imagine picking a 1 when any real number is equally likely.  What is the pdf?  Does this idea have a known use?  What is its name?
There could be a use for a uniform random real number.  It could end up always being integrated.  It could be a prior for a number that is completely unknown.  
It would be like a delta function, which integrates to a finite number though it doesn't have a well-defined value at zero.  The delta function is greater than any number at zero.   This uniform random on (-inf,inf) would be not well-defined anywhere.  It would be less than any positive number and greater than zero.  What is it?
 A: We can also approach the matter as follows: Let $\{X_{n}\}_{n\geq 1}$ be a sequence of real valued random variables following a uniform $U(-n,n)$ distribution. Does the sequence converge in distribution to a random variable?  
Convergence in distribution means
$$\lim_{n\rightarrow \infty}F_{X_n}(x) = F_X(x)$$
where the RHS is a distribution function, and the equality to hold for every $x$ for which  $F_X(x)$ is continuous.
Our distribution function is  
$$F_{X_n}(x)= \begin{cases}0&\text{if $x<-n$}\\\frac{x+n}{2n}&\text{if $x\in[-n,n]$} \\1&\text{if $x> n$.}\end{cases}$$
The first and the last branch are not defined as $n\rightarrow \infty$, since there is no $x$ lower than "minus infinity", or higher than "plus infinity". Then
$$\lim_{n\rightarrow \infty}F_{X_n}(x) = \lim_{n\rightarrow \infty}\Big (\frac{x}{2n} + \frac 12\Big) = 0+\frac 12$$
The constant function $1/2$ does not satisfy the properties of a distribution function, specifically
$$\lim_{x\rightarrow -\infty}\frac 12 =\frac 12 \neq 0,\;\;\lim_{x\rightarrow \infty}\frac 12 =\frac 12 \neq 1$$
The fundamental entity associated with a random variable is the cumulative distribution function, and not the probability density function. Since the limit of cdfs converges to a function that is not a cdf, the sequence does not converge to a random variable. Of course, as the comments indicated, this does not make it a useless concept, quite the contrary.
ADDENDUM Feb 1, 2014.
I found a nice piece of history related to this question: A.Rényi, in 
Rényi, A. (1955). On a new axiomatic theory of probability. Acta Mathematica Hungarica, 6(3), 285-335. motivates his axiomatic formulation of probability in terms of conditional probability by writing (p.285), among other things that 

"In the theory of KOLMOGOROV it has, for instance, no sense to speak
  about a probability distribution which is uniform on the whole real axis or
  on the whole plane, further it has no sense to say that we choose an integer
  in such a way that all integers (or all non-negative integers) are equiprobable."

