How do I output the smallest value entered into an array? (c++)

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Try to declare variables where they are used. Otherwise the code will be less readable.

The array rainfall is not yet initialized

double rainfall[ARRAY_SIZE];
//...
double highest = rainfall[0];
double lowest = rainfall[0];

So using its elements with indeterminate values for the variables highest and lowest does not make sense.

Declare and initialize the variables just before the loop where they are calculated.

double highest = rainfall[0];
double lowest = rainfall[0];

for(int x = 0; x < ARRAY_SIZE; x++)
{
    if(highest < rainfall[x])
    {
        highest = rainfall[x];  
    }
    if(lowest > rainfall[x])
    {
        lowest = rainfall[x];
    }
}

In this loop

for(int index = 0; index < ARRAY_SIZE; index++)
{
    cout << " Month " << index+1 << ": " ;
    cin >> rainfall[index];
    total_year += rainfall[index];

    if(rainfall[index] < 0)
    {
        cout << " Rainfall must equal to 0 or higher: " ;
        cin >> rainfall[index];
    }
}

move the statement

    total_year += rainfall[index];

after the if statement.

for(int index = 0; index < ARRAY_SIZE; index++)
{
    cout << " Month " << index+1 << ": " ;
    cin >> rainfall[index];

    if(rainfall[index] < 0)
    {
        cout << " Rainfall must equal to 0 or higher: " ;
        cin >> rainfall[index];
    }

    total_year += rainfall[index];
}

I would substitute the if statement for a while statement like

    while (rainfall[index] < 0)
    {
        cout << " Rainfall must equal to 0 or higher: " ;
        cin >> rainfall[index];
    }

but before using the variable total_year you have to initialize it

double total_year = 0.0;

The variable monthly_average is not used in the code. So its declaration can be removed.

Take into account that there are the following algorithms in C++ std::min_element, std::max_element, std::minmax_element that can be used to find minimum and maximum alements in an array or other container.

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