Down in the city's data portal, there's a collection of census data about the languages Chicagoans speak (or at least spoke, from 2006-2010). One of the measures is residents who "'speak English less than very well,' [categorized] by the non-English language spoken at home."
The Census measures 40 languages (including a category for "unspecified"), only one of which is not reported as spoken at home in Chicago (Scandinavian). Out of 418,320 such speakers, the top languages are Spanish (70 percent), followed by Polish (8.5 percent), Chinese (5.4 percent), Tagalog (1.6 percent), and Russian (1.2 percent). Given how much Ukrainian I hear around my neighborhood (even from teens), I was surprised to not see "Other Slavic" out of the top five.
The community area with the most English-as-a-second-language speakers? Belmont Cragin, with 31,142, of which some 25,000 are Spanish speakers and 4,000 are Polish speakers. Much of the northwest side is linguistically diverse; at a glance, West Ridge appears to be the most linguistically diverse.
The least linguistically diverse are south-side areas: the bottom 10 range from Chatham (169 non-native speakers) from West Garfield Park (10 non-native speakers, all Polish).
Spanish is the most widespread, of course, with a high of 47 percent in Gage Park, but Polish still hangs on in the northwest corner of the city (Dunning has 6,843 Polish speakers, the most of any community area).