It is definitely possible to get through huge numbers of levels using just a starter deck and the odd card purchased along the way using gold from wins. I've done it several times. Bit of a challenge, but certainly doable.
It may also interest you to know that you can make a killer deck with just common cards that holds up in PvP too.
We deliberately avoided making the rare cards just super-powered versions of common cards. A lot of them are of course strong, but those are the ones that cost 5 power points. Which is great until your opponent uses Lost, or Implode and has points left to spare....
Some of the rares are definitely more powerful but only in certain situations. That is what we aimed for.