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Someone in a previous thread asked about pushing Provia 100F, here's an un-scientific comparison of pushing 1 and 2 stops. These were all camera scanned with exactly the same settings and processing.
Under daylight they all look OK. 200 is barely distinguishable from 100 even on the light table, except maybe a tiny loss of shadow detail. At 400 the shadows are very light, more like a dark gray, and they have very little detail. There's a minor color shift from red towards yellow, but unless you're looking side-by-side you can't really tell. You also lose a little bit of saturation.
Under artificial light there's more difference. 200 shows a little bit of the color shift towards yellow, and at 400 it's very obvious. Under fluorescent the added green turns warm lighting an ugly puke green. The lack of shadow detail at 400 will also stand out more under dim lighting.
However in all cases, the color shift can be totally fixed in editing, and the light shadows can be darkened but you'll still lose detail. Personally I'd shoot at 200 without worry, and use 400 if I needed and just accept the slides will look ugly. Based on how 400 looks I wouldn't try pushing 3 stops to 800 (but it would be cool to see if anyone has tried).