Today is your lucky day.
I have found one record for a Jesse Dennis Burgess, presumably your father, and three records for a Jesse Burgess. While I am uncertain that Jesse Dennis Burgess and Jesse Burgess are the same individual, details in the four records seem to suggest that they are the same person. I found one record for a Dennis Burgess but I believe he is not the same individual, as he was described as a British citizen and sailing in a British ship. The information on Jesse Dennis Burgess and Jesse Burgess come from the website Ancestry.com (http://www.ancestry.com/), more commonly used for genealogical research, which has records of the arrival of passengers and ships' crews in the port of New York between 1820 and 1957. I found detailed sailing records of the ships on which he served at the ConvoyWeb website, http://www.convoyweb.org.uk/.
These are the records I found, in reverse chronological order.
(1) Jesse Dennis Burgess, age 21, signed onto the crew of the Liberty ship LEBARON RUSSELL BRIGGS (Frank Moore, master) on 14 Nov 1944, in New York City. The ship returned to New York on 29 March 1945, having sailed from Glasgow, Scotland, on 12 March. Burgess was a maintenance man on the ship and, at that point, had three years of sea-going experience. He was 5'7", 150 lbs., and had tattoos on both arms. Additional details below indicate that this is the voyage to Murmansk about which your father told you.
From records in ConvoyWeb, LEBARON RUSSELL BRIGGS sailed from New York to Liverpool in convoy HX 324, leaving New York on 4 December 1944, arriving in Liverpool 21 December. She was carrying general cargo plus railroad locomotives. The ship continued on to Murmansk in convoy JW 63, leaving Loch Ewe, Scotland, on 30 December 1944 and arriving in Kola Inlet (i.e., Murmansk) on 8 January 1945. She left Kola Inlet on 17 February 1945 in convoy RA 64 and arrived Loch Ewe on 28 February, then departed Liverpool on 13 March 1945, returning to New York on 29 March in convoy ON 290. So the ship was in Murmansk five or six weeks and was not necessarily iced in all winter, but it probably seemed much longer. See http://www.convoyweb.org.uk/ports/index.html?search.php?vessel=LEBARON%20RUSSELL%20BRIGGS~armain.
(2) A Jesse Burgess, age 21, signed onto the crew of the Liberty ship GEORGE GIPP (John Elizeus, master) on 14 June 1944, in New York City. The ship returned to New York on 28 September 1944, having sailed from Barry, Wales, on 11 September. Burgess was an A.B. (able seaman) with two years of sea experience. He was listed as 5'6 ½", 148 lbs., no mention of tattoos.
According to ConvoyWeb, GEORGE GIPP left New York 24 June 1944 in convoy HX 297. Most of the convoy was headed for Liverpool but GEORGE GIPP departed the convoy at some point and sailed to Oban, Scotland, arriving there 10 July. Between 11 July and 19 July she made several short-range voyages between various points in Scotland and England. Then between 1 August and 7 September she made several short runs from points in England to Seine Bay, France, the location of the Normandy invasion beaches. By 11 September she was in Barry, Wales, then on 12 September she was in Belfast, Ireland, from which she sailed in convoy ON 253 to New York, arriving 28 September. See http://www.convoyweb.org.uk/ports/index.html?search.php?vessel=GEORGE%20GIPP~armain.
(3) Jesse Burgess, age 20, signed onto the crew of the Liberty ship JOSHUA SENEY (Andrew G. King, master) in January 1944 in New York. The ship was scheduled to sail to an unspecified port on 29 January. Between 29 January and 9 February the ship was in Baltimore, and between 10 February and at least 12 February 1944 the ship was in Newport News, Virginia, scheduled to sail again on an unspecified date for an unspecified destination. The position Burgess held in the ship's crew is not identified in this record.
Records at ConvoyWeb (http://www.convoyweb.org.uk/ports/index.html?search.php?vessel=JOSHUA%20SENEY~armain), show JOSHUA SENEY leaving Hampton Roads (i.e., Newport News) on 13 February 1944 in convoy UGS 43, arriving in Oran, Algeria, on 3 March. Between 18 March and 8 April 1944 she was in several short-range convoys between various points in the Mediterranean, including Augusta, Sicily, and Naples, Italy. She departed from Augusta on 9 April 1944 in convoy GUS 36, returning to Hampton Roads on 2 May.
(4) Jesse Burgess, age 20, signed onto the crew of the Liberty ship OLIVER HAZARD PERRY on 13 May 1943 in New York. The ship returned to New York on 24 November 1943, having sailed from Bizerte, Tunisia, on 3 November. Burgess was an O.S. (ordinary seaman) with one year of experience. He was 5'6", 145 lbs., and had tattoos on both forearms.
On ConvoyWeb (http://www.convoyweb.org.uk/ports/index.html?search.php?vessel=OLIVER%20HAZARD%20PERRY~armain) are records of OLIVER HAZARD PERRY leaving Hampton Roads on 14 May 1943 in convoy UGS 8A, arriving in Oran, Algeria, on 3 June. It is not clear where she was (unless in Oran) between 3 June and 14 September, the date on which she sailed from Oran joining convoy UGS 16, arriving in Bizerta, Tunisia, on 17 September. There is another gap in the record until 28 October when she sailed from Augusta, Sicily, arriving again in Bizerta on 30 October. She left Bizerta on 4 November in convoy GUS 20 and returned to either Hampton Roads or New York on 24 November.
It is likely that your father sailed in other ships prior to 13 May 1943, since he supposedly had a year of sea experience by then. As next of kin you may be able to obtain more information on your father's merchant marine service by contacting the U.S. Coast Guard, which was and is responsible for issuing merchant marine documentation ("seaman's papers"). Write to this address:
U.S. Coast Guard National Maritime Center
Mariner Information (NMC-4)
100 Forbes Drive
Martinsburg, WV 25404
Telephone: (304) 433-3400
Submit as much identifying information as possible including name, date of birth, copy of death certificate if deceased, Social Security number, address, and Z or service number. Merchant mariner identification cards have serial numbers beginning with the letter Z, hence a "Z number."
Ron Carlson, Webmaster
Armed Guard website www.armed-guard.com