A lot of research seems to have been done on the challenges multilingualism poses for the human brain, but not so much on how much actually meeting those challenges improves one's overall cognitive capacity. At present the general approach seems to be summed up: "Cognitive science suggests that the brain has selective resources with limited capacity" (Emily Holmes et al.). Stanislas Dehaene in particular asks, "How can cerebral circuits that normally handle a single phonology, lexicon and syntax adapt to the storage of multiple language systems?", and suggests that "sophisticated mechanisms of segregation must exist to prevent cross-talk." These can be witnessed in action when "anyone trying the well-known party trick of reciting the number sequence while switching languages (for instance un, two, trois, four, cinq) will recognise that central coordination is a very plausible source of difficulty in translation and language switching." Source of difficulty maybe, but not yet apparently evidence of positive capacity for other tasks. Hopefully some work has gone down that road since.
Holmes, E. et al. "Can Playing the computer game 'Tetris' reduce the Build-Up of Flashbacks for Trauma? A proposal from Cognitive Science" Abstract.
Dehaene, S. "Fitting two languages into one brain" Brain (1999) 122(12)
Since this answer I have indeed found some reports on the beneficial cognitive effects of bi- if not multi-lingualism:
Researchers said the extra effort involved in using more than one language appeared to boost blood supply to the brain and ensure nerve connections remained healthy - two factors thought to help fight off dementia.
"We are pretty dazzled by the results," Professor Ellen Bialystok of Toronto's York University said in a statement.
"In the process of using two languages, you are engaging parts of your brain, parts of your mind that are active and need that kind of constant exercise and activity, and with that experience [it] stays more robust," she later told CTV television.
(Bilingualism delays onset of dementia
23:22 12 January 2007 by New Scientist and Reuters)
So his team compared the age dementia symptoms appeared in some 650 natives of Hyderabad, India, over a six-year period. About half spoke at least two languages. This group's symptoms started on average four and a half years later than those in monolingual individuals.
The same pattern appeared for Alzheimer's, frontotemporal and vascular dementia. The results also held true for a group of people who were illiterate, suggesting that the benefits of being bilingual don't depend on education (Neurology, DOI: 10.1212/01.wnl.0000436620.33155.a4).
A leading theory as to why bilingualism may affect dementia involves the constant suppression of one language, and switching between the two. "This permanent switching and suppressing offers you constant brain training," says Bak. ("Learn une autre langue to hold off dementia" New Scientist 9 November 2013.)