【Analysis】With weights of $1$, $2$, and $3$ grams, all weights from 1 to 6 grams can be measured.
The next weight is 8 grams, and it is found that 7 grams is skipped and cannot be measured,
Thus, with weights of $1$, $2$, $3$, and $8$ grams, all weights from 1 to 14 grams except 7 grams can be measured (1 to 6 grams can be measured with 1, 2, and 3 grams, and adding 8 grams to each of these measurements allows measuring 9 to 14 grams)
The next weight is 16 grams, and it is found that 15 grams is skipped and cannot be measured,
Thus, with weights of $1$, $2$, $3$, $8$, and $16$ grams, all weights from 1 to 30 grams except 7 grams, 15 grams, and 23 grams can be measured (1 to 6 grams and 8 to 14 grams can be measured with 1, 2, 3, and 8 grams, and adding 16 grams to each of these measurements allows measuring 17 to 22 grams and 24 to 30 grams, but 23 grams is missing because 7 grams is missing, so $7+16=23$ grams is missing)
The next weight is 32 grams, and it is found that 31 grams is skipped and cannot be measured,
Thus, with weights of $1$, $2$, $3$, $8$, $16$, and $32$ grams, all weights from 1 to 62 grams except 7 grams, 15 grams, 23 grams, 31 grams, 39 grams, 47 grams, and 55 grams can be measured
Therefore, the weights that cannot be measured are $7, 15, 23, 31, 39, 47, 55$ grams
$8 、$ Using 1g, 2g, 3g, 8g, 16g, 32g weights each (weights can only be placed on one side of the balance), the integer gram weight that cannot be measured, not exceeding 60g, is $\qquad$