There are $2019$ students sitting around circular table. Initially each of them have one candy. Teacher is allowed to pick one student, who has at least one can candy, and this student can decide, whether he gives his candy to his neighbour on the right or on the left. Prove that no matter what students teacher picks during the process, students can always ensure that any point of time no student has more than $2$ candies.
1. **Initial Setup**: 
   - There are 2019 students sitting around a circular table.
   - Each student initially has one candy.

2. **Process Description**:
   - The teacher can pick any student who has at least one candy.
   - The chosen student can give their candy to either their neighbor on the right or the left.

3. **Objective**:
   - Prove that no matter what students the teacher picks during the process, students can always ensure that at any point in time, no student has more than 2 candies.

4. **Strategy**:
   - Denote the students by $1, 2, \ldots, 2019$.
   - Each student initially has 1 candy.
   - We need to show that the number of candies any student has never exceeds 2.

5. **Key Insight**:
   - Consider the movement of candies. Each candy can be thought of as having an "identity" based on its initial position.
   - Let the $n$-th candy be the candy initially with the $n$-th student.

6. **Candy Movement**:
   - Each candy can only move to the neighboring students.
   - Specifically, the $n$-th candy can only be with the $n$-th student or the $(n+1)$-th student (mod 2019).

7. **Ensuring the Condition**:
   - At any point in time, the $n$-th candy can only be with the $n$-th student or the $(n+1)$-th student.
   - This means that the $n$-th student can have at most 2 candies: their own initial candy and the $(n-1)$-th candy (mod 2019).

8. **Conclusion**:
   - Since each student can only have their own candy and at most one additional candy from a neighbor, no student can have more than 2 candies at any point in time.

\[
\boxed{\text{No student can have more than 2 candies at any point in time.}}
\]