Guar Gum, also called Guaran or Guarkernmehl (German), is an important and widely available polymer that can turn detergent and water into an excellent "bubble juice". It is generally sold as a powder and in many countries (including the U.S.A.) is often available in local stores (unlike many other bubble-friendly polymers which need to be ordered on the Internet).
When used at the appropriate level, it can be very self-healing and produce giant bubbles even on a par with PEO. Guar-based juices can create bubbles that last considerably longer than PEO-based and HEC-based recipes.
See this recipe for an example of a recipe with easily found ingredients that can create giant bubbles in a wide variety of conditions. It benefits from the presence of either the baking soda/citric acid combination or baking powder. See the recipe for recommendations about hydrating the guar. It is a friendly and easy-to-mix polymer if you use an appropriate method -- see the recipe for tips.
Guar gum is a naturally occurring cellulose gum that is commonly used as a thickening agent in food products. It is similar (in some ways) to Xanthan Gum; although, Guar Gum seems to be more effective as a bubble juice primary polymer. (This conclusion is based on preliminary exploration -- it is quite possible that if the right amount of Xanthan were used that it would be comparable. The author, however, was not successful in replicating guar-based bubbles with xanthan.)
WHERE TO FIND IT Edit
Guar gum is generally pretty easy to find locally in the USA. Many health food stores and stores like Whole Foods have sections devoted to gluten-free baking -- or baking sections that include gluten-free ingredients. Look for guar gum there as it is least expensive when sold for this purpose. It is also often found where dietary supplements are sold in the digestive aids section as guar gum is very high in fiber. $5 of guar gum will make many many gallons of bubble juice.
If you cannot find it, ask the store manager. Bob's Red Mill is a common brand in the US. It is also available from a wide variety of suppliers on the internet, including the Bob's Red Mill website (but you must buy 8 - 8 ounce packages). At least some of the Sprouts Farmers Market stores, which are located in the region south and west of Colorado carry the Bob's Red Mill brand of guar gum as well. It can also be found online at some suppliers of craft supplies.
Different brands of Guar Gum may have different viscosities. It may take some experimentation to find what works best for any particular brand (or detergent:water ratio). As a primary polymer, in solutions with water:detergent ratio (with Dawn Pro or Dawn Ultra) on the order of 25:1, 0.5 grams to 3.0 grams per liter of water seems the optimal range. Guar bubbles can be very self-healing if you have enough guar gum in the solution. As a secondary or tertiary polymer Between 0.2 to 1.0 grams per liter of bubble juice seems to be recommended. A few people have reported that expensive industrial-grade guar gum works no better than the more common food-grade guar gum.
COMPATIBILITY: Many people have tried to combine PEO with Guar or Guar with Xanthan -- with which it has known to have synergies for viscosity building -- both of these combinations appear to result in mixes (based on multiple reports) that perform less well than with guar gum alone. There are quite a few positive reports using guar gum and minute amounts of PEO or PEO with minute amounts of guar gum, but it isn't clear how this compares with a guar-only mix with a slightly adjusted amount of guar. There have also been reports that guar gum + PEO + another polymer works better than any combination of just guar gum and PEO, but we haven't tested any such solutions.
SECONDARY/TERTIARY POLYMER: Prior to May 2012, guar gum was used mostly as a secondary or tertiary polymer in bubble juice to improve flow off of string wands. Some feel that it gives a glossy sheen to bubbles though I have been unable to distinguish between bubbles created with and without guar based on the sheen. It is also reported to improve the number of bubbles per dip that you get with a tri-string wand (though this has not been confirmed).
SUBSTITUTES: There aren't really any direct substitutes for guar gum. Some people find xanthan gum easier to find, but the juice created with xantham gum seems not to work as well as that made with guar gum. The xanthan seems to create bubbles that are more 'brittle' (for lack of a better word) and the amount must be adjusted.
Guar gum and water by themselves make a stable solution (although one with a limited shelf-life since guar gum and water by themselves are prone to biological degradation, and only keep for a few days unless preservatives are used). However, the addition of detergent often destabilizes the mix -- an effect that is exacerbated by baking soda or sodium citrate (or things like baking powder that contain them) as discussed here. As of March 2014, we do not have a recommended concentrated guar-based bubble juice, though the waterless concentrate with detergent, guar gum and baking powder detailed here works pretty well as long as you add more baking powder (or baking soda+citric acid) after diluting (the baking powder in the concentrate loses its ability to adjust the pH) though it is not a pretty concentrate. Development of a non-settling guar-based bubble juice concentrate will be an ongoing wiki project. If you have information on this topic, please let us know!
Miscellaneous Information Edit
WEIGHT FOR VOLUME: 1/2 packed teaspoon is about 2.1 grams.
Solubility and Mixing: Of interest: "... sodium and calcium chloride solution speed up guar gum hydration without affecting viscosity"  Guar gum is insoluble in isopropyl alcohol, propylene glycol, glycerine, and very cold water. Those liquids may be used to slurry guar gum to make mixing easier.
Other reading. An interesting article about guar gum is: http://journal.scconline.org/pdf/cc1971/cc022n01/p00043-p00060.pdf
This SBF post indicates that lab-grade guar gum does not behave better than the food-grade guar gum commonly used by bubblers.