Example 3 Set $A$ consists of integers, the smallest of which is 1, and the largest is 100. Except for 1, every element is equal to the sum of two numbers (which can be the same) in set $A$. Find the minimum number of elements in set $A$.

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Thought analysis: First construct a set $A$ that meets the conditions, showing that the minimum number of elements in $A$ cannot be greater than 9, and then further show that the minimum number of elements in $A$ is 9.

Solution: Construct a set with as few elements as possible to meet the conditions, such as: $\{1,2,3,5,10,20,25,50, 100\}$, then the minimum number of elements in set $A$ is no more than 9.

If $\left\{1,2, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, 100\right\}$ also meets the conditions, then $x_{1} \leqslant 4, x_{2} \leqslant 8, x_{3} \leqslant 16, x_{4} \leqslant 32, x_{5} \leqslant 64$.
But $x_{4}+x_{5}=96<100, \therefore x_{5}=50$,
$x_{3}+x_{4}=48<50, \therefore x_{4}=25$,
$x_{2}+x_{3}=24<25, \therefore x_{3}=\frac{25}{2}$, which contradicts that $x_{3}$ is an integer.

Therefore, the minimum number of elements in $A$ is 9.