First off - with anything health related, if you feel effects that are atypical and are concerned, get to a doctor. The advice we can provide here should in no way be considered medical advice or substitute for proper medical procedure.
It is impossible to say what the lowest dose is for any one individual - it depends on your personal sensitivity to it, how long an exposure (1 time, many times, over how long? (minutes, hours, days, weeks?)), exposure route (oral? inhalation?), age, gender, etc. Obviously incidental exposure when you are applying it will happen, but you aren't likely to panic and rush off to the hospital for that, but how much beyond that is impossible to say.
Values in the literature are usually expressed in terms of LD50 (lethal dose for 50% of population studied), but this is in a model animal (rat or mouse usually) and will be well above the dose at which you could get sick.
Having said that the CDC has a fairly comprehensive report (PDF) on the literature, with some human cases of all sorts of routes of exposure.
Based on the OP's comment to this post:
The OP is talking about an acute oral exposure. The lowest value with what is known as NOAEL (No Observed Adverse Effect Level) is 100 mg/kg/day for rats, dogs and rabbits in the CDC document linked above (see table 3-2, pp. 33-43). If you extrapolate that value to a human of 50 kg that is 5000 mg or 5 g per day. This equates to very close to 5 ml (1 teaspoon) of 100% DEET per day (density of DEET = 0.998 g/ml).
This NOAEL is for a variety of physiological systems: systemic, developmental, hepatic, reproductive and immunologic. Values for other systems were at least double the 100 mg/kg/day value discussed above.