The short answer: We don't know for certain because extant written accounts from the period are scarce.

The long answer: It is possible to make an educated guess.

The Globe Theatre was a wooden, twenty-sided, three-storey, open-air amphitheater that could (reportedly) house about 3,000 people at maximum capacity. The majority of that crowd would have been standing for the duration of the play, exposed to the fickle London weather. An excavation of the old Globe's foundations in 1988-89 revealed that the floor was covered in a layer of nutshells that helped to keep mud from the earthen floor under control, but the duration of a season would have been completely dependent upon the English climate. So we're probably talking about a performance window between late April and early October (the same as the modern Globe Theatre in London today). Records of winter productions by The Lord Chamberlain's Men/King's Men between 1599 and 1613 are all commissioned indoor performances at court or at private homes during weddings, holidays, etc.

It would have been common at the time for theatre companies to perform multiple plays in repertory. It is often presumed that this repertory was sequential and overlapping but it's possible that all the shows were in rotation throughout the season. There are mentions of Shakespeare's earlier plays from the 1590s being remounted at The Globe, but the constant demand for new material ensured that newer plays would have been favored.

The Globe was completed at some point after March of 1599. The most likely candidate for the first production in the new structure is Henry V. Henry V is easy to date thanks to a reference by the Chorus to the Earl of Essex's Irish expedition of 1599 that puts the time of its completion between March and September of that year. Just in time for a summer/fall premiere at the newly finished Globe.

[The opening Chorus to Henry V] (http://www.youtube.com/watch?v=CZFngXSDD0I) appears to reference the building in which it was intended to be performed:

... But pardon, and gentles all,

The flat unraised spirits that have dared

On this unworthy scaffold to bring forth

So great an object: can this cockpit hold

The vasty fields of France? or may we cram

Within this wooden O the very casques

That did affright the air at Agincourt?

However, although we know when it was written and can say that it was probably part of the 1599 season, the earliest confirmed performance of Henry V was in January of 1605. Dover Wilson, Regius Professor of English literature at the University of Edinburgh, suggests in the Cambridge New Shakespeare version that the first production mounted at The Globe may have been Julius Caesar. He notes a Swiss tourist's account of a performance of Julius Caesar on 21 September 1599 at the Globe. There is no mention of exactly how long the play had been running by that point

TL;DR:

• Multiple overlapping shows in repertory.

• Minimum run = ~3 weeks.

• Maximum run = ~24 weeks.

• Likely run = ~5-8 weeks. (My estimate, presuming a 3 - 6 show season.)

The 9 show run of A Game at Chess before the actors were prosecuted would have been considered very short indeed.