I've decided to explore Friedman's stuff better. I should start with n(k) because it's slowest widely known function defined by him.
Let's define the function L(n,m) as the number of Friedman strings which have n letters from alphabet {1,2,...,m}.
For example, let's compute L(4,2). First, we write all strings with n letters from alphabet {1,2,...,m}:
1111
1112
1121
1122
1211
1212
1221
1222
2111
2112
2121
2122
2211
2212
2221
2222
Then we remove strings written in bold text from this list because they're not Friedman. It contains 8 strings and so L(4,2) = 8.
Now we can redefine n(k) as the largest p so that L(p,k) > 0. The problem drops to explore the behavior of L(p,k) function.
For k = 2, we have the following sequence:
L(1,2) = 2
L(2,2) = 4
L(3,2) = 8
L(4,2) = 8
L(5,2) = 16
L(6,2) = 12
L(7,2) = 24
L(8,2) = 4
L(9,2) = 8
L(10,2) = 2
L(11,2) = 4
L(12,2) = 0
L(i,2) = 0 (for i > 12)
We can note that if m is even, then L(m+1,n) = L(m,n)*n. If m is odd, then it's not clear how L(m+1,n) compares to L(m,n). If we find that out, bounding n(k) might be straightforward.