How many Total Eclipse Saros cycles can run concurrently?
All solar eclipses occur in repeating cycles, or Saros Series, separated by 6585.32 days (approx 18y 11d and 8 hours. Thus any eclipse will be followed by another similar one approximately 18y 11d 8h later. The effect of the extra 8 hours, or 1/3 of a day, means that each successive eclipse in a given series will occur about 120 degrees further Westward on the Earth's surface, as the Earth rotates 1/3 of a complete rotation in those extra 8 hours.
A complete Saros Series can last anywhere between 703 years and 1064 years.
But not all solar eclipses are total - at any given time there will be Saros Series which are producing partial and annular as well as total eclipses. Currently (2015 CE* to 2054 CE) there are 40 different Saros Cycles in operation: Saros Series cycle numbers 117 to 156 inclusive. Saros Series 117 began in CE 792 and will end in CE 2054, so we are coming to the end of that Series. Saros Series 156 began in CE 2011 and will end in CE 3237, so we have just started that series. All Saros series begin and end with Partials, and have a varying number of Totals, Annulars and/or Hybrids** in the middle section.
But of these 40 series, only 15 are currently producing Totals (or hybrids), i.e. in the period CE2015 to CE2054.
And in the duration of the current Saros Series No 120 which included March 2015, there will 10 more totals, plus one hybrid, before the next one in Saros Series No 120, which occurs on 30 March 2033.
So taking the current time as typical, for Totals and Hybrids, there are about 10 or 11 OTHER Saros cycles running in parallel with any given Saros series.
*CE = Common Era, or the conventional way of numbering years without reference to Christianity, ie this year is CE 2015."
** A Hybrid is an eclipse which is partly total and partly annular at different points along its length