Big numbers: Ultrafactorials

Mathematica, 19 bytes

Sum[n!^n!,{n,0,#}]&

Apologies for the extremely clear code ;)

Jelly, 6 bytes

Try it online!

‘Ḷ!*`S // Main link: Argument n (integer)
‘      // Take n, increment by 1
 Ḷ     // Range from [0..n]
  !    // Calculates factorial for each [0..n]
   *`  // Raises each [0!..n!] to the power of itself
     S // Sum the resulting array

05AB1E, 5 bytes

Code:

Ý!DmO

Explanation:

Ý       # Take the range [0, ..., input]
 !      # Map factorial over each element
  Dm    # Exponentiate each element to itself
    O   # Take the sum

Uses the CP-1252 encoding. Try it online!