Having more than 1 or 2 alleles for a trait can greatly increase the number of phenotypes depending on the trait’s specific pattern of inheritance.
For example human blood type is controlled by 3 alleles (just referring to the ABO blood groups here): A B and O. A and B are both dominant to O but they are codominant to each other. So while you could have two people with different alleles like one is AA and one is Ao they would both have the same phenotype which is blood type A. In total there are four phenotypes here.
Again though the specific number of phenotypes will vary according to the inheritance pattern of the trait whether it is Mendelian incomplete dominance codominance polygenic etc.
Blood type chart: