an exceptional value
When a function takes values, most targets get hit constantly while one rare target might be neglected — never reached, or reached far too seldom. That neglected target is an exceptional value. It is the precise name for the 'one allowed exception' that runs through Picard's, Borel's, and Nevanlinna's theorems: the single value a function is permitted to treat differently from all the others. Identifying a function's exceptional value, if it has one, tells you the one place its otherwise democratic distribution of values breaks down.
There is a hierarchy of how exceptional a value can be, ordered from strongest to weakest. A Picard exceptional value is omitted entirely — the equation f(z) = w has no solutions at all (e.g. w = 0 for e^z). A Borel exceptional value is attained, but so sparsely that the counting function of its w-points grows strictly slower than the order rho. A Nevanlinna deficient value is attained, but the integrated counting function falls short of the maximal rate, measured by a deficiency delta(w) between 0 and 1; a fully omitted value has deficiency 1. Each notion is weaker than the last: omitted implies Borel-exceptional implies deficient. The theorems bound how many such values can exist: at most one Picard or Borel exceptional value; and Nevanlinna's defect relation caps the total deficiency, forcing the sum of all deficiencies to be at most 2.
Exceptional values are the heart of value-distribution theory: they are exactly the values where a function's behavior is special, and the great theorems are really statements about how few exceptional values are possible. The exponential is the clean model — 0 is its lone exceptional value at every level (omitted, Borel-exceptional, deficient with deficiency 1). A subtle point to keep clear: 'exceptional' is relative to the strength you mean. A value can be Nevanlinna-deficient without being omitted, so saying a value is 'exceptional' is incomplete until you specify Picard, Borel, or Nevanlinna — they are genuinely different thresholds.
The value 0 for f(z) = e^z is exceptional at all three levels: it is Picard exceptional (e^z is never 0), hence Borel exceptional (no w-points at all, so their counting function is identically 0, slower than the order 1), hence Nevanlinna deficient with deficiency delta(0) = 1. For e^z this is the unique exceptional value; every other value w is attained infinitely often with full density.
For e^z the value 0 is exceptional in every sense; all other values are attained with full density.
Always specify which kind: a Nevanlinna-deficient value need not be omitted, and a Borel-exceptional value need not be either; 'exceptional' is meaningless without the threshold. The counts differ too — at most one Picard/Borel exceptional value, but the deficiencies of deficient values sum to at most 2.