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I saw this spiral that had 'limited' out at 1 all the way around, looking like this. enter image description here

(Actually sorry that you must see my PC drawing skills)

If I could know what this is called and maybe a simple parametric equation to graph it, that would be great! Thanks

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    To help OP visualize the equations of the below answers, I've plotted them with Wolfram. This probably belongs here rather than as a child comment of only one of the answers. [Q's equations](http://www.wolframalpha.com/input/?i=Parametric+%7Bx(t)+%3D+(t%5E.5)cos(t),+y(t)+%3D+(t%5E.5)sin(t)%7D+with+4pi+%3E+t+%3E+0) and [Harry's equations](http://www.wolframalpha.com/input/?i=Parametric+%7Bx(t)+%3D+(1+-+e%5E(-t))cos(t),+y(t)+%3D+(1+-+e%5E(-t))sin(t)%7D+with+0+%3C+t+%3C+4pi) out to an interval of 4pi – 2017-02-03

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I don't know if there is an official name for these spirals, but I have called them limit cycle spirals when I discovered a family of such spirals in the past. These spirals derive from the gamma pulse, a function I developed for studying pulses in physical systems. It derives its name from the fact that it is the kernel of well-known gamma function. Thus,

$$\gamma(\tau;n)=\tau^ne^{-\tau} u(\tau)$$

and $$\int_0^\infty \gamma(\tau;n) d\tau=\Gamma(n+1)$$

where $u(\tau)$ is the Heaviside step function and $\tau$ is a dimensionless time.

Now, if $n\in\Bbb{R}^+$, $\gamma(\tau;n)$ is an ordinary pulse with a characteristic rise time and pulse-width. However, if $n\in\Bbb{C}$, you can get some really interesting curves in the complex plane. More specifically, if $n$ is purely imaginary, then you get the family of limit cycle spirals. The figure below shows several examples of limit cycle spirals created with the gamma pulse. As an added bonus, if you plot both $\gamma$ and $-\gamma$ you get some interesting yin-yang plots. (Here, we define a yin-yang curve as one that divides the disc into two congruent perfect sets; no more, no less.)

You can find more information on the gamma pulse in The Compleat Gamma Pulse and the linked PDF manuscript therein.

enter image description here

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    I haven't checked back on this for a while, but great answer. Thanks! – 2017-03-28
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I'm not sure if it has a name, but the parametric equations $x(t)=(1-e^{-t}) \cos(t)$, $y(t)=(1-e^{-t}) \sin(t)$ for $0 \leq t$ model what you drew.

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    Are you sure? The plot seems to escape. http://www.wolframalpha.com/input/?i=Parametric+%7Bx(t)+%3D+(1+-+e%5E(-t))cos(t),+y(t)+%3D+(1+-+e%5E(-t))sin(t)%7D – 2017-02-03
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    Good catch, I didn't specify my interval. $t$ must be greater than 0. – 2017-02-03
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    With some bounds on that, it ends up looking like this: http://www.wolframalpha.com/input/?i=Parametric+%7Bx(t)+%3D+(1+-+e%5E(-t))cos(t),+y(t)+%3D+(1+-+e%5E(-t))sin(t)%7D+with+2pi+%3E+t+%3E+0 – 2017-02-03
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    Looks great! Why e^-t? _To make it look like an e?_ Haha, just kidding. – 2017-02-04
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    Thanks! In general, a nice form for this curve would look like $x(t)= f(t) \cdot \cos(t)$, $y(t)= f(t) \cdot \sin(t)$ for some interval of $t$, say $[a, \infty]$. This, paired with the conditions that $f(a)=0$ and $f(x) \to 1$ as $x \to \infty$, would model your curve well. For my set of equations, I chose $f(t)=1-e^{-t}$. Q the Platypus chose $f(t)=\frac{t^{2}}{t^{2}+1}$. – 2017-02-04
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It looks like a modified Archimedean spiral. $$ x(t) = \frac{t^2}{t^2 + 1} \cos(t) \\ y(t) = \frac{t^2}{t^2 + 1} \sin(t) $$

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    [Q's equations](http://www.wolframalpha.com/input/?i=Parametric+%7Bx(t)+%3D+(t%5E.5)cos(t),+y(t)+%3D+(t%5E.5)sin(t)%7D+with+4pi+%3E+t+%3E+0) and [Harry's equations](http://www.wolframalpha.com/input/?i=Parametric+%7Bx(t)+%3D+(1+-+e%5E(-t))cos(t),+y(t)+%3D+(1+-+e%5E(-t))sin(t)%7D+with+0+%3C+t+%3C+4pi) out to an interval of 4pi – 2017-02-03
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    @Axoren I've made an improvement to the spiral which makes it more like the desired one. – 2017-02-03
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Hatcher mentions it in his notes on introductory point-set topology as being given by the polar equation $$r = \frac{\theta}{\theta+1}\ \text{ for } \theta \ge 0$$

but he doesn't give it a name (nor does he call the so-called "topologist's sine curve" by its name in this same chapter). In the notes, the graph together with the unit circle is an example of a space that's connected but not path-connected.

I know you asked for a parametric equation, but the polar one is quite nice and specifically mentioned in the reference.

Here's its graph through Desmos -- yours isn't bad for a Paint job!

Plot from Desmos

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    Looks just right! Thanks. I'll work on my drawing skills too. – 2017-02-03