Spirals are used as design guides and elements or paths for text, shapes, and particles. Here's a few resources for using and creating spirals in After Effects and other Adobe applications.
Think outside the box (or rather inside the sphere) with A 5-Part Exploration Of After Effects Spiral Filters by Brian Maffitt -- the grand daddy of spiral tutorials (pictured). This is from an era before the Path Text plug-in and the Textacy text engine!
3D Type QuickTip by Ko Maruyama presents some similar ideas, but with some new approaches available in CS3. The finished project file (tornado_3d) is still available.
James Chiny offers a free Golden Ratio preset for After Effects at HYPOLY. It generates an overlay of a Golden spiral and rectangles using a Stroke effect. Be sure to check out his other downloadables, like his Ellipse Tool Plus. See also Lightroom's Golden Ratio & Spiral guides, an AE Portal post with some related resources.
Taming the Spiral at AE Portal shared a few references for creating spirals in Illustrator. Also interesting is How to Create Sheet Rolls with Custom Spirals and Quick Tip: How to Replace the Spine in Illustrator Blends on VectorTuts+.
See AE Help for the Animation preset list with several spiral options.
After Effects: Spiral Text by John Dickinson is a free project that "uses a simple index expression to offset the text layers and nulls to control the distribution using Expression Control sliders. This allows them to be freely distributed with one null while setting keyframes for Y-Position and Rotation with another."
Using Motion Sketch to build a Spiral Path in 3D Space by Fang Xiang does what it says.
Tornado Text, a video tutorial by Eran Stern, shows how to use per-character 3D text animation to animate text along a path in the shape of a 3D tornado.
The Wonderful World Of Shapes In After Effects here on Aetuts+ includes some spirals.
Among many similar examples is Animated Swirls in Adobe After Effects by Steve Holmes at Layers.
Spirograph designs are a popular use of expression; see Spiral shaped motion paths, Spirograph-style pattern generator in AE by Chris Zwar (inspired by the spirograph tutorial from Ayato), and Dan Ebberts on spirograph expressions. Along those same lines but using Particular and expressions is Generate Your Own Splendid Spirographical Designs by Christopher Kirkman on Aetuts+.
Variations on spirograph patterns can also be made in After Effects with standard tools. Tutorials by Satya Meka (on AEtuts) and others were referenced in A few spirographs in After Effects. Several projects by Tom Beddard can also be leveraged: Guilloché Patterns, Harmonograph, and especially the Pixel Bender filters Fractal Explorer and Droste Effect. Carl Larsen even has an After Effects tutorial for the Droste Effect. Omino Spindala, another free Pixel Bender filter, has spiral controls. There's even an iPad/iPhone app that creates similar graphics, Vectoria.
Ewan Smith's video for Aetuts+, QuickTip – Creating Circular And Spiral Motion Paths, used a light, a simple expression, and Trapcode Particular. Motion Graphics Exchange wrote out the expression for you in Creating Circular And Spiral Motion Paths.
Send A Happy 2011 New Year Card uses Trapcode Particular to create a spiral Christmas tree, on Aetuts+ originally Dec 30, 2009. A similar result is produced by ParkourProduction on Youtube in After Effects Tutorials-Spiral Particles.
Create a “Light-tastic” Logo Reveal Animation by Lloyd Alvarez [thanks] is a 2-part video at Aetuts+ that uses expressions and Particular to create a simple spiral.
Peder Norrby shows you how to use Trapcode Form to make a spiral shape in Form Spiral - Quick Tutorial. Also, Mylenium provides some free spiral projects for Form (this also) and Digieffect Delirium plug-ins.
There was an example tornado (a 3D spiral) that shipped with an early version of the CC Particle World plug-in, and it really boiled down to changing just 2 or 3 parameters from the default. It may still be in the demos from Cycore. It was close to what Andrew Kramer started in his Energy tutorial. SuzTV has a preset that may also get you started in Particular, Tornado preset for Particular 2 in After Effects.