I use Aspirin regularly for the reason Camille stated. There is a good evidence that aspirin water can boost the plant immune system. Also, it helps plants under drought condition. SA is a kind of stress hormone (among many other hormonal function), so when I see stressful situation (early infection, deflasking, recently received plants), I spray/drench with 1-4 tablets (325mg per tablet) of cheap, uncoated aspirin per gallon. As with any preventative measures, it is probably difficult to notice the effects, but there is a good scientific evidence to support it in other plants. It also reduces the chance of viral infection for certain types of virus. Here is a good review (it may be abstract-only):
Salicylic Acid, a multifaceted hormone to combat disease. - PubMed - NCBI
Camille, I didn't know that SA had influenced sap sucking insects. Interesting. Also if you happen to have a citation stating that there is a counter-effect against neurotrophic pathogen (I'm guessing that you are talking about fungal pathogen?), I appreciate it. I thought that induction of systemic acquired resistance (SAR) involves killing off the infected tissues, so isn't it supposed to be effective to neurotrophic pathogens?
To Ray's question, ASA seems to be mimic the effect of SA in several plant reaction (induced immunity, reduction in ethylene production etc).
fishmom, I think that you can get the same effect by using vinegar. The pH seems to be the key rather than aspirin. This was mentioned in this paper:
:RASKIN, I (1992) ROLE OF SALICYLIC-ACID IN PLANTS.
ANNUAL REVIEW OF PLANT PHYSIOLOGY AND PLANT MOLECULAR BIOLOGY
Volume: 43 Pages: 439-463
I personally haven't seen any negative effects, but I wouldn't use it continuously. The continuously raised immunity states might reduce the growth (my speculation based on the belief that there is a reason why SAR is not raised all the time in the normal state). But I have seen a couple papers showing that applying SA frequently can results in bigger plants.
Also there are some evidences showing that some plants use SA for allelopathy. In other words, the roots excrete SA to prevent other plants to spread their roots (to win the competition). If this is the case, SA may inhibit root growth in some plants.
As a related note, aspirin does reduce the germination rate of seeds (I have done this experiment), so it has the opposite effect of gibberelins.