The exponential factorial is defined by the recurrence relation
 |
(1)
|
where 
and 
is a positive integer. The first few terms are
therefore
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
... (OEIS A049384). The term 
has 183231 decimal
digits.
The exponential factorial is therefore a kind of "factorial power tower," with exponentiation associated to the right.
The sum of the reciprocals of the exponential factorials is given by
 |  |  |
(6)
|
 |  |  |
(7)
|
(OEIS A080219). This sum is a Liouville number and is therefore transcendental.
