It is expected that there are infinitely many primes for which it is $1$; indeed one expects this to hold for more than 70 per cent of all primes. However, this is open.
The exact value is only know for very few primes; I think only up to $67$ (this was true until some years ago, but perhaps I missed something).
There are however example where one know that it is somehow large, in particular larger than $p$. The first example for this is due to Cornell and Washington.
Recall that the Vandiver conjecture states it is not divisible by $p$, so that this size threshold is particularly interesting.
For more information (in particular details on the above heuristic) you could look at Schoof "Class numbers of real cyclotomic fields of prime conductor" Math Comp. (2003); this should be freely available.