I've been trying to get every trophy in Super Smash Bros. for 3DS without resorting to buying them from the Trophy Shop. I've played flawless Trophy Rush games and various other modes over 400 times now (not exaggerating) and I've still not gotten any new trophies. I have about 530 trophies right now, but still need around 70 more to beat the 600 trophy challenge.
I know there are more trophies to collect since they appear on the Trophy Shop, but I only want to start getting them through there until I'm certain there's no other way to get them.
Best Answer
No, you cannot get every trophy without buying any from the shop.
There are eight trophies that are only available from the trophy shop. These trophies represent the fighters that are included as costume swaps of other fighters:
These trophies cannot initially appear in the shop and will only appear after completing Classic mode with the character who they are a costume swap of (Olimar for Alph and Bowser Jr. for the Koopalings).
All other trophies can be earned by other means.
Mario Wiki has a great list that includes unlock criteria for all 685 non-DLC trophies as well as the growing list of DLC trophies here.