A baker makes 6 batches of muffins. There are 23 muffins in each batch. A box can hold 5 muffins. What is the greatest number of full boxes the baker can package?

The baker makes a total of 6 x 23 = <<6*23=138>>138 muffins.

So, he can put 138/5 = <<138/5=27.6>>27.6 boxes of muffins.
Therefore, the greatest number of full boxes the baker can package is 27. Answer: \boxed{27}.