Actually the spammer simply increases her fees. One must keep in mind that the spammer relies the fact that only a very small portion of the spam transactions will actually get mined regardless of the fees because of the fixed blocksize. A dynamic blocksize with a proper fee structure can be a very effective deterrent against this type of spam attack because by threatening to mine the spam it imposes a very significant risk to the spammer.
The spammer increasing their fees just causes legitimate transaction fees to increase since wallets use dynamic fee calculators, this prices the spammer out of the blocks since transaction fees are relatively small for the legitimate users but large for the spammer(since they are trying to buy a lot of block space). Without a cap fees would likely be too low to price the spammer out of the blocks.