What's a good cheap flash for taking repeating flash photographs using a Nikon D3100?
I' not sure exactly what this is called, so I'm not sure how to ask the question do I hope this makes sense.
I think you're looking for a stroboscopic mode, where the flash fires multiple times in one exposure.
The Nikon SB-800 and SB-900 have this feature — Nikon calls it "repeating mode". (The SB-700 does not.) That probably doesn't fit your definition of "cheap", though.
Another option is the Sigma EF610 DG Super, which you can get for $225, or the Nissin Di866 for around $300. These third-party flashes are reverse-engineered "dedicated flashes", which come in specific versions for Nikon, Canon, Pentax, etc., and have advanced features rivaling the system-branded models.
Metz has a stroboscopic mode too, but only on their $350+ model. Also, the cheaper Sigma EF610DG ST (presumably for "standard", as opposed to "super") and the lower-end Nissin models don't have the feature.
The Promaster 7500EDF is another $250 option, but I hesitate to recommend it, because a) it has a more-limited version of the feature with a max of 10 flashes in a sequence and b) for Pentax at least Promaster has done a poor job keeping up compatibility with new camera bodies, and while they might be better with Nikon I wouldn't count on it.
I don't know of any cheap flashes that offer stroboscopic mode. The high end OEM flashes have it - Canon 580 and Nikon SB-800 & SB-900. Canon calls it Multi Mode, Nikon calls it Repeating Flash.
Your other option is to get cheap flashes, use them at low power and use Pocket Wizard MultiMaxs, they offer the option to pop the flash multiple times in one exposure. This might be the best option, because then you have radio triggers and a flash instead of just a flash.
Yongnuo has (at least) two models for Nikon that have what they call "multi mode"
The YN-468 that costs about $90 and the YN-565EX for about $150 (from eBay, new, including shipping)
I did some research on flashes. Im not a pro or living on profits from photography. I have the Nikon D5200 with an assortment of lenses. I wanted the Nikon SB-910 but it was way out of my justifiable price range for a hobby. However, I found a clone of the SB called a MK-910 by Meike for $103 which is lot cheaper than the $550 option. I ordered the package deal with the mk-rc10n wireless flash triggers for $132.99 and thought that was a great deal. Units flash more flashes or strobes or whatever more than I'll ever need. In the last couple days since i received them I've been impressed with the quality of both the flashes and the remotes, has already made for some awesome pics! To elaborate on what i meant by "more than ill ever need". Yes, it does have repeating flash mode. The settings range is easily adjustable from 2 to 24flashes at 1hz to 100hz. The build quality is very solid. The display and menu can be a little confusing if you have never used a flash before (thats me, the rest of you will do fine), but after a few outings it all starts to come together as rather simple. The mk-910 is not as powerful as the sb-910 (so ive heard), but it is stronger than the sb-800, so i suppose that makes it a cheap flash that would satisfy some, but not all. Ty