Sure, I can help you create a test case generator for the given problem statement. Here's the test case generator function for the "Mobile Computing" problem:

```python
import random

def generate_random_weight():
    return random.randint(1, 1000)

def generate_random_room_width():
    return round(random.uniform(0.00001, 10.0 - 0.00001), 8)

def tcgen():
    num_datasets = random.randint(1, 10)  # Number of datasets to be generated
    datasets = []
    
    for _ in range(num_datasets):
        r = generate_random_room_width()
        s = random.randint(1, 6)  # Number of stones
        weights = [generate_random_weight() for _ in range(s)]
        datasets.append((r, s, weights))
    
    return datasets

def format_test_case(datasets):
    test_case_str = f"{len(datasets)}\n"
    for r, s, weights in datasets:
        test_case_str += f"{r}\n{s}\n"
        test_case_str += "\n".join(map(str, weights)) + "\n"
    return test_case_str.strip()

# Generate test cases
datasets = tcgen()
test_case_str = format_test_case(datasets)
print(test_case_str)
```

### Explanation:

1. **`generate_random_weight` Function**: Generates a random weight for a stone between 1 and 1000.
2. **`generate_random_room_width` Function**: Generates a random room width between very close to 0 and just below 10, formatted to 8 decimal places for precision.
3. **`tcgen` Function**: 
   - Generates a random number of datasets (between 1 and 10 for testing purposes; you can adjust this range as needed).
   - For each dataset, it generates:
     - `r`: A random room width.
     - `s`: A random number of stones (between 1 and 6).
     - `weights`: A list of random weights for the stones.
   - Appends each dataset to a list of datasets.
4. **`format_test_case` Function**: Formats the generated datasets into a string that matches the input format specified in the problem statement.

### Sample Output:

The `print(test_case_str)` will display the generated test cases in the format expected by the problem statement. Here's an example of what the output might look like:

```
3
5.67890123
4
123
456
789
101
2.34567890
5
234
567
890
123
456
9.87654321
1
999
```

This generator will create varied test cases to help you thoroughly test your solution to the problem.