Clustered standard errors (My research question is economics based, but for simplicity, I'm using a non-economics example)

Suppose I'm trying to find out whether average body temperature of a population is equal to 37 degrees Celsius. I take a random sample of people and take their body temperature. 

The problem is, some people have taken their body temperature once, but others have taken their temperature multiple times (twice, thrice, or even ten times).

Ordinarily I would regress body temperature on a constant, and do hypothesis testing with $H_0: \beta_0 = 37$. However, I am worried about serial correlation. 

Would clustering the standard errors by person be sufficient to correct for this serial correlation? (e.g. using reg bodytemperature, cluster(person) in Stata)
I am not sure this will answer your question, but let's me try with a concrete example using Stata and its 1978 Automobile Data in order to regress car's price on mileage (mpg).

sysuse auto, clear    
reg price mpg, robust


This gives a price estimate of -238.9, with a standard error (s.e) of 57.5. Then, if I run the same regression after duplicating the observations three times.

expand 3
reg price mpg, robust


I get as expected a much lower s.e. of 32.9. Finally, clustering the s.e. by make (a Car Id) 

reg price mpg, cluster(make)


gives a s.e. of 57.2 and makes the trick. 

However, if you expand twice some observations and thrice others (as in your example). Then, you will get a larger s.e but also a different estimate. But, again, clustering by make reduces the s.e. Here is the code:

sysuse auto, clear
expand 3 if price&gt;4000 &amp; price&lt;=6000
expand 2 if price&gt;6000
reg price mpg, robust
reg price mpg, cluster(make)


Note that the price and mpg variables have been expanded without any change for each car. In your example, if the ones who have taken their temperature multiple times have each time a different temperature, then you may consider introduction an individual fixed effect.