AppleScript 64 bit but no big integers?

Largest number which I can declare
(eg set anInteger to someBigNumber as Integer)
as an integer is 2^31-1.
No big integers in AppleScript?

No, AppleScript integers are signed 32-bit. If you do calculations that result in bigger values, it automatically rolls them over into reals. Of course they lose enough precision to be accurate well before 64 bits.