Common folklore dictates that the Surreals were discovered by John Conway as a lark while studying game theory in the early 1970's, and popularized by Donald Knuth in his 1974 novella.
Wikipedia disagrees; the page claims that Norman Alling gave a construction in 1962 which modified the notion of Hahn series fields to realize a field structure on Hausdorff's $\eta_\alpha$-sets, and that this construction yields the Surreals when considered over all ordinals -- the article pretty explicitly characterizes Conway's construction as a popularized rediscovery more than 10 years later (!).
The given reference links to a paper from 1962 (received by editors in 1960) which seems to fit the bill, however this paper gives no indication that Alling considered his construction extended to all ordinals nor does it contain mention of proper classes that I can find.
Is there any evidence indicating that Alling considered a proper class sized version of his construction almost a decade before Conway did?
Alling published a book in 1987 which constructs the Surreals in exactly this manner*, but this is (of course) more than a decade after Conway's construction was popularized. Was this construction known to Alling earlier and only written up in book form when he realized there was a more widespread interest, or were proper-class considerations in this arena genuinely not in play before Conway?
*This is incorrect, thanks to Philip Ehrlich for pointing out my mistake. Alling's 1987 book doesn't construct the surreals as modified formal power series in the fashion of his 1962 paper; the methods involved are more subtle. On January 3, 1983 and December 20, 1982, Norman Alling and Philip Ehrlich respectively submitted papers which constructed isomorphic copies of the surreals using modified versions of the constructions in Allings 1962 paper, and these submissions eventually lead to the publication of two papers to this effect. The 1987 book mentioned above contains expansions of these works in sections 4.02 and 4.03, respectively. (see Philip's excellent answer below for a better description of events)