Well, PEF (peak expiratory flow) varies thorough the day and it's at its lowest in the morning (which is when patients are usually instructed to take it x3, before their meds), so taking the highest value balances that out a little.
Also, we are measuring PEF i.e. maximum speed of expiration, not average speed of expiration ;) And a patient's personal best (the highest PEF reading they've recorded for a span of 2-3 weeks) is useful for determining asthma zones (green for when a patient's PEF measures 80%-100% of their best value, yellow for 50%-80% & red for <50%), which in turn are needed to make an asthma action plan so that the patient knows when they can treat their symptoms without going to the hospital and when they should. Using the highest reading also gives a little leeway in such cases as ≤80% PEF is not as bad as it could be!
Personal best PEF is also used to determine if a patient's meds need to be changed or the dose increased/reduced, etc.
References:
Reddel HK, Marks GB, Jenkins CR. When can personal best peak flow be determined for asthma action plans? Thorax 2004; 59: 922-4.
Asthma action plan. American Lung Association.
Bailey W, Gerald L. Peak expiratory flow rate monitoring in asthma.