If you want to produce a complete gzip-compatible binary string, with the header etc, you could use gzip.GzipFile together with StringIO:

try:
    from StringIO import StringIO  # Python 2.7
except ImportError:
    from io import StringIO  # Python 3.x
import gzip
out = StringIO()
with gzip.GzipFile(fileobj=out, mode="w") as f:
  f.write("This is mike number one, isn't this a lot of fun?")
out.getvalue()

# returns '\x1f\x8b\x08\x00\xbd\xbe\xe8N\x02\xff\x0b\xc9\xc8,V\x00\xa2\xdc\xcc\xecT\x85\xbc\xd2\xdc\xa4\xd4"\x85\xfc\xbcT\x1d\xa0X\x9ez\x89B\tH:Q!\'\xbfD!?M!\xad4\xcf\x1e\x00w\xd4\xea\xf41\x00\x00\x00'

There is a module gzip. Usage:

Example of how to create a compressed GZIP file:

import gzip
content = b"Lots of content here"
f = gzip.open('/home/joe/file.txt.gz', 'wb')
f.write(content)
f.close()

Example of how to GZIP compress an existing file:

import gzip
f_in = open('/home/joe/file.txt')
f_out = gzip.open('/home/joe/file.txt.gz', 'wb')
f_out.writelines(f_in)
f_out.close()
f_in.close()

EDIT:

Jace Browning's answer using with in Python >= 2.7 is obviously more terse and readable, so my second snippet would (and should) look like:

import gzip
with open('/home/joe/file.txt', 'rb') as f_in, gzip.open('/home/joe/file.txt.gz', 'wb') as f_out:
    f_out.writelines(f_in)