This is the only solution possible for using nothing but cut:
echo "s.t.r.i.n.g." | cut -d'.' -f2- [repeat_following_part_forever_or_until_out_of_memory:] | cut -d'.' -f2-
Using this solution, the number of fields can indeed be unknown and vary from time to time. However as line length must not exceed LINE_MAX characters or fields, including the new-line character, then an arbitrary number of fields can never be part as a real condition of this solution.
Yes, a very silly solution but the only one that meets the criterias I think.