I'm not sure why all these answers didn't quite meet par as some of them contained correct information, so I'll share what I know and dump a list of sources from a paper I wrote a few years back in grad school (for a 1 credit class). Basically, if treated right, the caffeine in plants like coffee and tea is shelf-stable for quite some time. In earlier centuries when coffee and tea drinking spread to places like Europe, travel was slow but steamed and dried tea leaves or dried or roasted coffee seeds could retain their stimulating effect on long ocean voyages. The major active compounds in Khat/Chat (Catha edulis), are cathinone and cathine (nurspeudoephedrine). The former, however, breaks down rapidly to cathine after picking the leaves (1-3 days or so, supposedly), however, and the plant loses much of its stimulating effect. Whereas caffeine can be imbibe rather rapidly as coffee or tea as well, khat must be chewed for quite some time to get even a mild effect (I can say this from personal experience working in East Africa for my PhD) equivalent to a cup of coffee. So, while coffee/tea and khat can be grown in tropics around the world, khat isn't particularly well-suited to long-distance travel, particularly prior to rapid trans-national shipping when tastes for stimulating beverages in places like Europe and the Americas were first taking off. However, I've heard from a toxicologist friend from California that khat is grown there as an ornamental. It's technically since become illegal under US law to grow the plant due to its placement on our drug scheduling laws as an analog to ephedrine, but it's such a uncommon commodity and unremarkable looking shrub that I think unless you were growing it for re-sale harvest, it wouldn't register on most people's radar.

Some sources (and apologies for the crap formatting as I cut and paste from a word doc):

Anderson, David, Susan Beckerleg, Degol Hailu and Axel Klein

2007 The Khat Controversy: Stimulating the Debate on Drugs. Oxford: Berg.

BBC

2009 Ethiopia Clamps Down on Khat Dens. 8 January. Uncredited article.

http://news.bbc.co.uk/2/hi/africa/7817355.stm. Accessed 30 Oct. 2009.

Fluckiger and Gerock

1887 Contributions to the Knowledge of Catha Leaves. Pharmaceutical Journal of

Transvaal. 18: 221.

Garowe

2009 Baidoa Protests Against Al Shabaab Khat Order. 25 March. Uncredited article.

http://allafrica.com/stories/200903260469.html. Accessed 30 Oct. 2009.

Gebissa, Ezekiel

2004 Leaf of Allah: Khat & Agricultural Transformation in Harerge, Ethiopia 1875-

1991. Oxford: James Currey Ltd.

Halbach, H.

1972 Medical Aspects of the Chewing of Khat Leaves. Bulletin of the World Health

Organization. 44: 21-29.

Jones, Aiden

2009 More Somali Migrants Say Britain Should Ban Khat. The Christian Science

Monitor. 8 May. http://www.csmonitor.com/2009/0508/p06s04-woeu.html.

Accessed 20 Nov. 2009.

Kennedy, J. G.

1987 The Flower of Paradise: the Institutionalized Use of the Drug Qat in Northern

Yemen. Dordrecht, Germany: Reidel Publishing.

Mercier, Jaques

1982 Un Mythe Ethiopien d’Origine du Café et du Kat. In Abbay: Etudes Ethiopiennes

1980-1982 – Cahier N. 11. Jean Chavaillon, ed. Pp. 143-180. Paris: Centre National

de la Recherche Scientifique.

Pendell, Dale

2002 Pharmako/Dynamis. Berkeley: North Atlantic Books.

Reveri, Raman

1983 Catha edulis Forssk.: Geographical Dispersal, Botanical, Ecological and

Agronomical Aspects with Special Reference to Yemen Arab Republic. Gottingen.

Robson, N. and Sebsebe Demissew

1989 Celastraceae. In Flora of Ethiopia, Vol. 3. Egziabher, Tewolde, et al. eds. Pp.

331-347. Addis Ababa: Addis Ababa University Press.

Varisco, Daniel Martin

2004 The Elixer of Life or the Devil’s Cud? In Drug Use and Cultural Context

‘Beyond the West.’ Ross Coomber and Nigel South, eds. Pp. 101-118. London: Free

Association Books.