Compute the number of geometric sequences of length $3$ where each number is a positive integer no larger than $10$.
To solve this problem, we need to find all possible geometric sequences of length 3 where each term is a positive integer no larger than 10. A geometric sequence is defined by the property that the ratio between consecutive terms is constant. Let's denote the terms of the sequence by \(a, ar, ar^2\), where \(a\) is the first term and \(r\) is the common ratio.

1. **Identify the possible values for \(a\) and \(r\):**
   - \(a\) must be a positive integer between 1 and 10.
   - \(r\) must be a positive integer such that \(ar\) and \(ar^2\) are also integers between 1 and 10.

2. **Check each possible value of \(a\) and find valid \(r\):**
   - For \(a = 1\):
     - \(r = 1\): Sequence is \((1, 1, 1)\)
     - \(r = 2\): Sequence is \((1, 2, 4)\)
     - \(r = 3\): Sequence is \((1, 3, 9)\)
   - For \(a = 2\):
     - \(r = 1\): Sequence is \((2, 2, 2)\)
     - \(r = 2\): Sequence is \((2, 4, 8)\)
   - For \(a = 3\):
     - \(r = 1\): Sequence is \((3, 3, 3)\)
   - For \(a = 4\):
     - \(r = 1\): Sequence is \((4, 4, 4)\)
   - For \(a = 5\):
     - \(r = 1\): Sequence is \((5, 5, 5)\)
   - For \(a = 6\):
     - \(r = 1\): Sequence is \((6, 6, 6)\)
   - For \(a = 7\):
     - \(r = 1\): Sequence is \((7, 7, 7)\)
   - For \(a = 8\):
     - \(r = 1\): Sequence is \((8, 8, 8)\)
   - For \(a = 9\):
     - \(r = 1\): Sequence is \((9, 9, 9)\)
   - For \(a = 10\):
     - \(r = 1\): Sequence is \((10, 10, 10)\)

3. **Count the valid sequences:**
   - Sequences with \(r = 1\): \((1, 1, 1)\), \((2, 2, 2)\), \((3, 3, 3)\), \((4, 4, 4)\), \((5, 5, 5)\), \((6, 6, 6)\), \((7, 7, 7)\), \((8, 8, 8)\), \((9, 9, 9)\), \((10, 10, 10)\) (10 sequences)
   - Sequences with \(r = 2\): \((1, 2, 4)\), \((2, 4, 8)\) (2 sequences)
   - Sequences with \(r = 3\): \((1, 3, 9)\) (1 sequence)

4. **Total number of sequences:**
   - Adding all valid sequences: \(10 + 2 + 1 = 13\)

The final answer is \(\boxed{13}\)