Here's some GAP code which exhaustively enumerates all nested primes. It's a backtracking algorithm, adding a new digit at each step. It the current number is a prime, it prints it out, otherwise, throws it away.
DigitsToInt:=function(d)
return Sum([1..Size(d)],i->10^(Size(d)-i)*d[i]);
end;;
NestPrime:=function(d)
local i,k;
for i in [1,3,7,9] do
d:=Concatenation(d,[i]);
k:=DigitsToInt(d);
if(IsPrimeInt(k)) then
Print(k,"\n");
NestPrime(d);
fi;
d:=List([1..Size(d)-1],j->d[j]);
od;
end;;
for d in [[2],[3],[5],[7]] do
k:=DigitsToInt(d);
Print(k,"\n");
NestPrime(d);
od;
Note that GAP's IsPrimeInt is a deterministic primality test for $n \leq 10^{13}$, which is sufficient in this case.
Which outputs:
2
23
233
2333
23333
23339
2339
23399
233993
2339933
23399339
239
2393
2399
23993
239933
2399333
29
293
2939
29399
293999
2939999
29399999
3
31
311
3119
31193
313
3137
31379
317
37
373
3733
37337
373379
3733799
37337999
37339
373393
3739
37397
379
3793
3797
5
53
59
593
5939
59393
593933
5939333
59393339
59399
593993
599
7
71
719
7193
71933
719333
73
733
7331
7333
73331
739
7393
73939
739391
7393913
73939133
739393
7393931
7393933
739397
739399
79
797
So, yes there is a largest nested prime, and it's 73939133 (in agreement with Ross Millikan's answer and Sloane's A024770).