If you want a deodorant that works very well, saves a lot of money, and is much better for the environment and you, I strongly recommend the Thai Deodorant Stick.
Please note that it is primarily a deodorant and not that good an antiperspirant. If you need an excellent antiperspirant, use something else. However most people do not need a high-powered antiperspirant every day. For instance, if I'm headed into a situation where it is critical that I stay dry (like a tense presentation in front of many people under hot lights), then yes I still use a good antiperspirant such as Dry Idea Roll On. However 99% of the time (even for most presentations -- and I give a lot of presentations) a good deodorant is all I need every day. Plus sweating is a good thing. It keeps you cooler.
To use the Thai Deodorant Stick effectively, it is best to wet it slightly and rub it back and forth on your underarms making sure to cover them thoroughly. This takes a little more time than typical liquid underarm deodorants/antiperspirants that spread themselves out even after application. However, these are much more costly for both you and the environment.
Saves money: Dry Idea Roll On works extremely well as both a deodorant and antiperspirant and costs about $5 for 3.25 ounces (good for about a month of use). That's about $60/year. Whereas a $7 Thai Stick (4.25 ounces) will last about 2 years (based on my use) for about $3.50/year.
Less waste: almost all of the Thai Stick is product you use -- it is a solid mineral crystal -- so there is little waste afterwards, just the cover and the base. Whereas other deodorants leave you with many more containers made of much more plastic as a waste product in the end.
Fewer allergy problems: more people seem to react to aluminum chlorohydrate, aluminium zirconium tetrachlorohydrex (or pentachlorohydrex), or the long list of other chemicals that usually are in other commercial deodorants. Alum (the mineral salt that makes up the Thai Stick) seems to cause fewer problems.
Less damage to clothing: the Thai Stick is a water soluble salt that does not seem to leave stains. It is made of alum -- see: http://en.wikipedia.org/wiki/Alum and http://www.deodorantstones.com/faqs.html. Dry Idea can leave stains behind after repeated use -- especially on my T-shirts, just as described at: http://en.wikipedia.org/wiki/Aluminium_zirconium_tetrachlorohydrex_gly. Some other deodorants may leave white marks (see: http://www.consumerreports.org/cro/home-garden/beauty-personal-care/personal-care/antiperspirant-claims-10-07/overview/anti-ov.htm)
Caveats: The crystal in the Thai Stick can chip/crack if dropped on a hard surface. If it breaks, the resulting hard edges may discourage use. Since the stick lasts so long, this risk of breakage may be significant for some. [Update: we have dropped, cracked, and broken 2 of these now so I would say the lifespan is more like 1 year rather than 2]
Aluminum: sorry to pop some bubbles but there is aluminum in the Thai Stick: KAl(SO4)2•12H2O. However it contains less aluminum than competing products such as aluminum chlorohydrate and aluminium zirconium tetrachlorohydrex and in a form that seems less harmful.
For additional background reading with numerous references see: http://en.wikipedia.org/wiki/Deodorant.
(BTW, it's interesting that the price for things that perform really well with good reviews tend to go up in price on Amazon. I bought mine for $5.76 each but now these are about $7 each. Regardless it is still a good deal.)
Update March 21, 2017: I noticed that even this deodorant started causing a rash and itching for me. However, now that I am retired, I found I really don't need a deodorant most of the time -- further reducing cost and effort. Now I occasionally (once a week or less) use a very small amount of a variety of deodorants only when really needed so that I can avoid rashes and keep my resource usage to a minimum.