First consider the polynomial ring $R=F[x,y]$ for a field $F$. One chain of prime ideals in this ring is $(0)\subseteq (x)\subseteq (x,y)$. Now $(x,y)$ is a maximal ideal, but there are other maximal ideals, for example $(x+1,y)$.
The easiest way to eliminate the other maximal ideals is to pass to the localization at the prime $M=(x,y)$ so that the new ring $R_M$ is a local ring. It is a property of localization that the prime ideals contained in $M$ will have prime counterparts in $R_M$, with the same containment relations and everything. Thus the chain $(0)_M\subseteq (x)_M\subseteq (x,y)_M$ will be a properly ascending chain of prime ideals in the new ring $R_M$.
Even if you are not handy with localization now, you probably will need to be soon, so trying to understand this type of example is worthwhile.