Bigger, badder competitors (i.e. VISA) seem like they haven't been interested in competing with Square, and you can leverage momentum & recognition against smaller newcomers.
The last thing VISA wants to do, is take over the entire payment system. That's why they're not really attempting to compete with Square or PayPal. They could trivially buy someone like Square.
VISA's golden goose is exactly where they're sitting today: no serious anti-trust burden, massive margins, low overhead brand-based business model. The minute they start trying to own everything, the goose gets shot. Today they do $5.4b in profit on just $12.7b in sales - what can a very modest business like Square offer them on top of their massive 42% net income margins to offset the anti-trust scrutiny they'd be taking on? Absolutely nothing.
If VISA tries to take over the payment processor space, Discover and Mastercard will immediately go in for the kill on anti-trust.
Square has more recognition with the public at large - but that also has little pressure on merchant adoption. After all, you're not going to refuse to swipe your card at the coffee shop because their PoS isn't Square.
Anecdotally over the past couple of years I've seen a proliferation of other PoS systems (all iPad-based) at merchants around me that are decidedly not Square. Square may have carved out the market initially, but the vacuum is being rapidly filled with players that aren't Square.
Square more ancillary offerings are really terrible though, like their online store and loins. They are also offering more services with "hidden" fees like instant deposit which they charge an extra fee if you read the fine print. They are also losing out in online transactions with google, amazon, visa, mastercard, and amex doing half ass jobs.
The B&M marketplace is in for a big shift with the chip and pin starting to commence. I've only used it twice, but so far the machines that read them are ridiculously slow and cumbersome. I'm still waiting on my Square reader to see if they have a good solution.
This is a big pie, and Square needs to capitalize on its good start if not be overuse by the old goliaths.