The function increments the ASCII value of the first character by one, replaces every 'e' with two 'd's, and keeps the rest of the string unchanged.

[CODE]:
```python
def approx_function(s):
    result = chr(ord(s[0]) + 1) + s[1:].replace('e', 'dd')
    return result
```

Let's verify the behavior with some additional inputs:

COMMAND: PYTHON(./temp/function.py "example" "replace" "side" "test" "text")