It really depends on both the drug and the person. But on the basis of physical addiction, it is possible after the first try. The only drug that comes to mind for this is methamphetamine which is arguably the worst drug to be addicted to due to its long lasting effects (Up to 3 days). Heroin (diamorphine) on the other hand rarely does result in instant addiction, this is shown in normal community pharmacy with such medicines as co-codamol being sold all the time (Codine is essentially converted to morphine by the cytochrome p450s within the liver, al be it at a 1:12 potency).
Physical addiction is a result of dopamine within the shell of the nucleus accumbens (Pleasure centre). By using addictive drugs your providing an external source of dopamine release (for heroin, this is via opiate receptors), your body will then get used to external release and deactivate its natural methods. By stopping you will then have decreased dopamine levels resulting withdrawal symptoms. This works in a similar way to how prednisolone works and how stopping it will result in withdrawal side effects. This is a slow process that needs to be reinforced to happen.
