```python
def rate_cap(price: int, rate: int, most: int) -> int:
    """
    Raises a price by a whole-percent rate, but never by more than a stated largest increase.
    Returns the new price, with the increase rounded down. A price raised by nothing comes back as it was.
    
    :param price: The original price.
    :param rate: The percentage increase rate.
    :param most: The maximum allowed increase.
    :return: The new price after applying the rate, capped by the most parameter.
    """
    if rate < 0:
        raise ValueError("Rate below zero is not supported.")
    
    # Calculate the potential increase
    potential_increase = price * (rate / 100)
    
    # Cap the increase by the most parameter
    increase = min(potential_increase, most)
    
    # Apply the increase to the price
    new_price = price + int(increase)
    
    return new_price
```