Problem 86

Let's denote the 5 micro-videos as A$_{1}$-A$_{5}$. We start by considering a scenario with only 2 videos, A$_{1}$ and A$_{2}$:

- If the number of likes for A$_{1}$ > the number of likes for A$_{2}$, and
- The expert score for A$_{2}$ > the expert score for A$_{1}$,
then we can have at most 2 excellent videos.

Next, we consider a scenario with 3 videos, A$_{1}$, A$_{2}$, and A$_{3}$:

- If the number of likes for A$_{1}$ > the number of likes for A$_{2}$ > the number of likes for A$_{3}$, and
- The expert score for A$_{3}$ > the expert score for A$_{2}$ > the expert score for A$_{1}$,
then we can have at most 3 excellent videos.

By extending this pattern, we can infer that among these 5 micro-videos, we can have at most 5 excellent videos.

So, the answer is: $\boxed{5}$.

This problem tests our understanding of basic logical reasoning, making it a fundamental question.