Glad to see some other magazine fans! FYI, I did a quick scan and these are the complete (or fairly complete) sets I found at first glance:
The Strand (first half of 1893)
Punch (most of 1914 and 1920, bits of other years)
Graham's Magazine (all of 1848)
The Arena (much of 1891)
Atlantic Monthly (1859-62, 4 complete; 1863-4 missing 1 month each, bits of other years)
Blackwood's Edinburgh Magazine (All of 1843 and 1844, bits of other years)
Continental Monthly (all of 1862 and 1864, bits of other years)
Lippincott's Magazine (many, but each year is missing some)
I am not sure how many of these are fairly stand-alone and how many heavily rely on being read in sequence, but I think I will start with the ones which are complete years. S/he who dies with the largest library in Calibre wins, right?
ETA: Link to
Bookshelf for easy browsing