Tickets went on general sale at 10am, with thousands held in an online queue as servers struggled to cope with demand.
By midday there were almost 30,000 waiting to purchase tickets to the production, which is directed by Lyndsey Turner and runs from 6 August to 31 October 2015.
Those who get through to the booking stage are limited to six tickets per household across the whole run, and photo ID of the 'lead booker' will be required to gain admission into the theatre.
But those who miss out today shouldn't lose heart completely – a number of £10 tickets, available for every performance, will go on sale next year.
Many have taken to Twitter to share their experiences and vent frustrations:
Trying to book tickets for Benedict Cumberbatch in Hamlet. Number of people in queue ahead of me? 16,343.
— Simon Mayo (@simonmayo) August 11, 2014
What an absolute brouhaha; all for one ticket as well… Still, going to see Cumberbatch as Hamlet. Be jel
— Doug Cairns (@DougCairns) August 11, 2014
I don’t even like Benedict Cumberbatch that much yet am in a virtual queue of 16677 people to get tickets to see him in Hamlet #hamletbatch
— Elizabeth (@elizmase) August 11, 2014
My position in the queue for Cumberbatch Hamlet tickets? 27,552. pic.twitter.com/UOVOhu41MH
— Michael W (@MagichaeI) August 11, 2014
If you're happy to pay a booking fee to beat the queue, click here for tickets