MySQL Sub Sub Query seems to loop

A commenter mentioned that it's hard to understand what you're trying to do here. I agree. But I will take the risk of trying to puzzle it out.

As usual with this sort of query, it's helpful to take advantage of the structured part of structured query language, and try to build this up piece by piece. That's the secret to creating complex queries that actually do what you want them to do.

Your innermost query is this:

          SELECT tblclient.ClientID
            FROM tblclient
            LEFT JOIN tblenquiry ON tblclient.EnquiryID = tblenquiry.EnquiryID
            WHERE NOT tblclient.EnquiryID IS NULL
             AND YEAR(EnquiryDate) = 2014
             AND MONTH(EnquiryDate) = 9

It is saying, "give me the list of ClientID values which have enquiries in September 2014. There's a more efficient way to do this:

          SELECT DISTINCT tblclient.ClientID
            FROM tblclient
            JOIN tblenquiry ON tblclient.EnquiryID = tblenquiry.EnquiryID
           WHERE tblenquiry.EnquiryDate >= '2014-09-01'
             AND tblenquiry.EnquiryDate <  '2014-09-01' + INTERVAL 1 MONTH

Two changes here: First, the NOT ... IS NULL search is unnecessary because if the item you're searching on is null, there's no way for your EnquiryDate to be valid. So we just change the LEFT JOIN to an ordinary inner JOIN and get rid of the otherwise expensive NULL scan.

Second, we recast the date matching as a range scan, so it can use an index on tbl.EnquiryDate.

Cool.

Next, we have this query level.

  SELECT MIN(tblinvoice.InvoiceID) AS InvoiceID
    FROM tblinvoice
   WHERE tblinvoice.ClientID IN (
            /* that list of Client IDs from the innermost query */
            )
  GROUP BY tblinvoice.ClientID

That is pretty straightforward. But MySQL isn't too swift with IN () clauses, so let's recast it in the form of a JOIN as follows:

  SELECT MIN(tblinvoice.InvoiceID) AS InvoiceID
    FROM tblinvoice
    JOIN (
           /* that list of Client IDs from the innermost query */
         ) AS inq ON tblinvoice.ClientID = inq.ClientID
  GROUP BY tblinvoice.ClientID

This gets us the list of invoice IDs which were the subject of the first enquiry of the month on behalf of each distinct ClientID. (It's hard for me to figure out the business meaning of this, but I don't understand your business.)

Finally, we come to your outermost query. We can also recast that as a JOIN, like so.

SELECT SUM(InvoiceProductTotal)
  FROM tblinvoiceproduct
  JOIN (
        /* that list of first-in-month invoices */
       ) AS inq2 ON tblinvoiceproduct.InvoiceID = inq2.InvoiceID

So, this all expands to:

SELECT SUM(InvoiceProductTotal)
  FROM tblinvoiceproduct
  JOIN (
      SELECT MIN(tblinvoice.InvoiceID) AS InvoiceID
        FROM tblinvoice
        JOIN (
               SELECT DISTINCT tblclient.ClientID
                 FROM tblclient
                 JOIN tblenquiry ON tblclient.EnquiryID = tblenquiry.EnquiryID
                WHERE tblenquiry.EnquiryDate >= '2014-09-01'
                  AND tblenquiry.EnquiryDate <  '2014-09-01' + INTERVAL 1 MONTH
             ) AS inq ON tblinvoice.ClientID = inq.ClientID
       GROUP BY tblinvoice.ClientID
       ) AS inq2 ON tblinvoiceproduct.InvoiceID = inq2.InvoiceID

That should do the trick for you. In summary, the big optimizing changes are

1. using a date range scan.
2. eliminating the NOT ... IS NULL criterion.
3. recasting your IN clauses as JOIN clauses.

The next step will be to create useful indexes. A compound index (EnquiryDate, EnquiryID) on your tblenquiry is very likely to help a lot. But to be sure you'll need to do some EXPLAIN analysis.