Besides English, what language(s) are most spoken world-wide? While Mandarin Chinese has the most speakers overall, you're not likely to find a Chinese speaker in Lesotho or Bolivia or Serbia or Uzbekistan. I'm guessing French would be the next most universal, but I just don't know.
It is no measure of health to be well adjusted to a profoundly sick society. -J. Krishnamurti
Table 3 on this Ethnologue page lists languages by number of speakers. The first ten are various Chinese languages, English, Arabic, Hindi, Bengali, Portuguese, Russian, Japanese, German.
I was surprised not to see Spanish in your list but I see it's just a mistake. Spanish is, as I thought, second if you count all the Chinese ones as one.
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There are several imponderables here. As already mentioned, Chinese and Spanish are the most spoken natively. However, as you say, in many places you really rely on people who have a second or other language. French is possibly useful as it, like English and Spanish, comes from a former colonial power who may still have some connections in the area, even if only negative. Because of the size of the former USSR, Russian might be useful in that area, although less so worldwide.
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Here's a source whiich answers the question several ways. If you add secondary to primary language spoken, French comes in 5th, after Mandarin, English, Spanish and Russian. By numbers of countries where French is spoken(where it is native, official, or widely used in trade, tourism, minority, preference), French comes in second only to English.
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Now we're getting somewhere! If I went on a Friends of Tuva excursion http://www.fotuva.org/ I'd best know Russian; if I fell from the sky over Rwanda and survived, I could get along in English or French. It appears that if one knows Mandarin, English, Spanish, French, Russian, and Arabic, one can go pretty near anywhere and find someone to talk to.
It is no measure of health to be well adjusted to a profoundly sick society. -J. Krishnamurti