There are two cases I can think of where multiple blurs would be performed in succession on a single image.
First, when performing a large-radius blur, it may reduce the total computation if you first downsample the image (which is a blur) and then perform a smaller-radius blur on the downsampled image. For example, downsampling an image by 4x and then performing a 10px-wide Gaussian blur on the result would approximate performing a 40px-wide Gaussian blur on the original—but is likely to be significantly faster due to improved locality in sampling and fewer samples taken overall.
The initial downsampling filter is often simply a box (as shown above), but it can also itself be something more sophisticated, such as a triangle or bicubic filter, in order to improve the approximation.
This is a Mitchell-Netravali (cubic) downsample followed by a Gaussian. Interestingly, it turns out that using a Gaussian for the initial downsampling doesn't make such a great approximation if your goal is to use it to produce a larger Gaussian.
An initial downsampling step is also frequently used when implementing visual effects like depth of field and motion blur, for similar reasons.
A second reason to perform multiple Gaussian blurs is to approximate a non-separable filter by blending between various Gaussians of different radii. This is commonly used in bloom, for example. The standard bloom effect works by first thresholding to extract bright objects from the image, then creating several blurred copies of the bright objects (usually using the downsample-then-blur technique just discussed), and finally weighting and summing them together. This allows artists a greater level of control over the final shape and appearance of the bloom.
Here, for example, is a weighted sum of three Gaussians (red line) which produces a shape that's more narrowly peaked and heavier-tailed than a single Gaussian (blue line). This is a popular sort of configuration to use for bloom, as the combination of a narrow, bright center with a wide, diffuse halo is visually appealing. But since this kind of filter shape isn't separable, it's cheaper to make it out of a mixture of Gaussians than to try to filter with it directly.
Another variation on this idea is the concept of a diffusion profile used with subsurface scattering for skin rendering. Different blur radii may be used for the red, green, and blue channels to approximate the way different wavelengths of light scatter differently, as in GPU Gems 3 skin shading chapter by Eugene d'Eon and Dave Luebke. In fact, that paper uses a mixture of seven different Gaussians, with different R, G, and B weights for each, to approximate the complicated non-separable, wavelength-dependent scattering response of human skin.