let $M$ be a closed ortientable irreducible 3-mfd, let $T$ be a non-separating torus in $M$, we cut $M$ along $T$ and glue two solid tori along the two boundary tori, we get a new closed 3-mfd $M_1$ (we need $M_1$ also is irreducible). Now If there exists nonseperable torus $T_1$ in $M_1$, we go on the above process, we get a new closed 3-mfd $M_2$ ...
My question is, whether you can find a $M$, choose suitable $T_i$, glue solid tori suitablely, the process will go infinitely?
or can you prove that it is impossible to find such an example? (for example, from $M$---> $M_1$, some invariant of 3-mfds decrease strictly).
After Kevin's example, I added the condition "$M_1$ also is irreducible". This condition is natural in the original field (for this question): 3-mfd with Anosov flow.