Shakespeare was certainly very popular and successful in his lifetime, and 1606 in is a great year for Shakespeare. Given your framing plus his fame and familiarity, I'm going to answer this question a bit from Shakespeare's point of view. James Shapiro (a huge Shakespeare scholar)'s book 1606: The Year of Lear is a great examination of the year for Shakespeare. Not to mention, the year probably also gave us Macbeth and other fan-favorites were certainly in his company, The King's Men's, repertoire at the time. You're right to mention Marlowe as a big figure in English renaissance drama, but his death in 1593 puts him a bit out of the purview of the year in question. His plays might have still been performed in 1606 as plays were the property of the companies they were written for, not the playwrights themselves.

In 1606, Shakespeare found himself at a weird point in his career. He started off in the early 1590s with a string of successful comedies and histories, two staple genres. The Taming of the Shrew, Love's Labours Lost, and A Midsummer Night's Dream were a few of those comedies and the histories were his first tetralogy, Henry IV 1+2, Henry V, and Richard III. Shakespeare shifted around the turn of the century into a tragic mode: Romeo and Juliet, Julius Caesar, Hamlet. Around the time of James I's accession to the throne, things shifted on him big-time. This began the era of the tragicomedy, a more complex genre that usually had a twisty-turny plot with plenty of dark moments but a happy ending after all. Shakespeare tried his hand at these later in his career and produced some great (and some difficult) works: Antony and Cleopatra, Pericles, Winter's Tale.

The Revenger's Tragedy by Thomas Kyd was a thrilling play that premiered around then about lust and murder and deception. It's pretty thrilling and was definitely at the time, too. Ben Jonson, the English Poet-Laureate (a title he held over Shakespeare) had just released his exciting comedy, Volpone, about the seedy criminal underbelly of Venice. 1606 was around the conclusion of the so-called War of the Theatres. This was a kind of a three-way battle between Ben Jonson (the established and well-practiced poet) against John Marston and Thomas Dekker (who Jonson saw as upstarts with poor taste). Jonson wrote his play, Poetaster , (a word Jonson invented that means something like Poet-Pretender) to lampoon Dekker and Marston. In 1606, Marston's play Parasitaster offered some response. You could see that one (it's not so good). Around 1606 a play called Roaring Girl premiered, telling the story of London celebrity Mary Frith, usually called Moll Cutpurse, who was a cross-dressing criminal. That play was by another two great dramatists, Thomas Middleton and Thomas Dekker. Thomas Heywood, a playwright who claimed to have had "a hand or at least a maine finger" in more than 200 plays had his play If You Know Not Me, You Know Nobody performed in 1606.

I list a lot of plays in the previous paragraph to emphasize my answer to one of your questions. Shakespeare was popular, yes. But Shakespeare was far far faaaar from dominant. A tiny minority of plays from the period were printed (Heywood, the prolific guy I mentioned before, only had 20 of his 200+ printed), and still, we have more than 600 play texts from the period. Some scholars suggest that more than 4,000 plays were performed on the English stage from 1580-1640. Of them, Shakespeare wrote less than 40. Another author wrote more than half of those 2,000: his name was Anonymous. Shakespeare was not the poet laureate of England in his lifetime and he was not the most commercially successful, either. He didn't write enough to be. We recognize him today as the great writer of the language probably because he is, but also probably because we have a lot of his stuff, still. In 1623, Shakespeare's works were compiled by two actors who knew him, Hemmings and Condell, and published as the first folio. Without this, about half of his plays would not exist today, including The Tempest and Macbeth.

All that is to say that there was a lot for you to see in London in 1606. Everything I mentioned and dozens more plays being performed everyday (except Sunday), by dozens of playwrights and playing companies in a bunch of playhouses dotted around London.