Yes, it is possible, and called transaction spamming. If you consider it cost effective spamming probably depends on how much it is worth to you to bloat the blockchain, already at a size measured in double-digit GBs, for everybody who keeps a complete copy!
However, there are some possible mitigations already implemented. First of all, many clients delay forwarding very small transactions and delay incorporating them into blocks or even require a transaction fee for them. As the transaction spam problem becomes larger, surely more mining pools will stop bothering unless a transaction fee is included.
Finally it is not strictly necessary for every single client to keep every single block of the blockchain around. Hence the long-term impact of such spamming is probably rather limited, and its widespread implementation, if not now then at least sooner or later, costly.