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If there are three different alleles to a gene, is it possible that the first is dominant to the second, but recessive to the third?

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Sure it is!

When we say that an allele is dominant on another what we really mean is that the phenotype obtained from that specific mutation of the gene somehow "masks" the phenotype of the other.

The easiest example would be that of pigment. A gene X could code for an enzyme which produces a red pigment. A mutation in X could result in a allelic variant x coding for a non-functional enzyme that does not produce pigment. In this case X will be dominant on x, as in the presence of both the pigment will still be produced (because you will have half functional and half non-functional enzyme molecules).

Things can, however be slightly complicated.

For instance, if the allele R codes gives a red pigment and r gives a white pigment Rr could give pink. This is called incomplete dominance.

On the other hand you could have co-dominance, where WW is white, ww is black and Ww is black and white.

The example you are asking for is called serial dominance and the "classical" example given is that of the rabbit coat color where you have for alleles with a dominance hierarchy like:

c+ > cch > ch > c

c+ gives a fully functional enzyme, called tyrosinase
c gives a non-functional enzyme, resulting in an albino phenotype
cch (chincilla phenotype) and ch (hymalayan phenotype) give partly functional enzyme (interestingly, the latter gives a temperature sensitive fur color).

In reality the rabbit coat genetics is even more complex, see this documents (PDF) for a more complete list of all the various loci and possible alleles involved.

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