Q:

What is the upper limit on gravitational time dilation? 

First, please forgive me: I'm not versed enough in mathematics to even approach working this out for myself. 
Now, the question: how much "slower" could time pass for a person around a supermassive black hole (such as S5 0014+81)?  Would this increase indefinitely with the mass of the hole, or is there something about maximum density or whatnot that places a limit on this?  
I'd imagine that the increasing volume of a hole would somewhat cancel the effects of increasing mass, but I'm such a layperson that I'm merely speculating at this level of knowledge. 

A:

The time dilation becomes infinite at the event horizon.  That is, an observer watching from far away would see the rate that a clock runs slow to zero as the clock approaches the event horizon.  This is not directly related to the mass of the black hole.  The mass determines where the event horizon is, but it's the distance relative to the event horizon that determines the time dilation. 
For a static black hole the time dilation is given by:
$$ \frac{d\tau}{dt} = \sqrt{1 - \frac{r_s}{r}} \tag{1} $$
where $r_s$ is the event horizon radius and $r$ is the distance from the black hole.  For interest, the event horizon radius is given by:
$$ r_s = \frac{2GM}{c^2} $$
Anyhow, when you are far from the black hole $r \gg r_s$ so $r_s/r \ll 1$ and the time dilation is small.  As $r$ decreases towards $r_s$ the time dilation gets greater until when $r = r_s$ equation (1) gives us:
$$ \frac{d\tau}{dt} = \sqrt{1 - \frac{r_s}{r_s}} = \sqrt{1 - 1} = 0 $$
so at $r = r_s$ time stops completely.