if the r th term is Tr,
Tr=2^[1+2^(r-1)]
take log to the base e on both sides
Sn=T1.T2.T3...............Tn
ln|Sn| = ln|T1| + ln|T2|+ ln|T3| + ........+ln|Tn|
ln|Sn| = [1+2^0].ln|2| + [1+2^1].ln|2| + [1+2^2].ln|2| ...... +[1+2^(n-1)].ln|2|
ln|Sn| = {n + [1+2+4+8+......2^(n-1)] }.ln|2|
[1+2+4+8+......2^(n-1)] = 1.(2^n-1)/(2-1)=2^n-1
ln|Sn| = {n + 2^n-1 }.ln|2|
ln|Sn| = [2^n+n-1].ln|2|
taking anitln of both sides
Sn=2^[2^n+n-1]
tests,

S1=2^[2]=4 (the first term)
S2=2^[5]=32=4.8 (the product of first two terms)
S3=2^[10]=1024=4.8.32 (the product of first, second and third terms)
... and so on