Is there a binary tree structure with fast access to recently accessed elements and worst $O \left( \log n \right )$ complexity?
The amortized cost of entering an element to splay tree is $O(\log n)$ so in the worst case $n$ operations would do $O(n\log n)$ steps. There is a conjecture about splay tree that says that it is as optimal as a binary search tree up to constant factor called Dynamic optimality conjecture.

Tango trees are online, self adjacent trees that achive $O(\log \log n)$ competitive ratio compared to offline optimal binary search tree which is the best known.