Corporal James Dunsby, 31, died yesterday in hospital where he had been in a serious condition after falling ill during the exercise on July 13, one of the hottest days of the year.
His death follows that of colleagues Edward John Maher and Lance Corporal Craig John Roberts, who also lost their lives after collapsing during the military exercise on south Wales' highest mountain.
Corporal James Dunsby
In a statement issued through the Ministry of Defence (MoD) today the family of Cpl Dunsby, who was a member of the Army Reserves (The Royal Yeomanry), said the 31-year-old had the "most infectious enthusiasm for life".
"He was the most loving and dependable husband, not to mention the most handsome of men of whom could not be more cherished," they said.
"James was and will remain a dearly loved, son, brother and husband. He had the uniquely wonderful ability to endear, enchant and captivate all who he met with his naughty sense of humour and highly intelligent wit.
"With James as your friend you were ensured loyalty, strength, allegiance, protection and most of all a damn good laugh.
"James adored the army and believed so passionately in his duty as a protector of Queen and country and of the realm.
"James was so dearly loved by so many and will be sorely missed. Hurrah and Huzzah for James Dunsby; a greater man you could not meet.
"We ask to be allowed to come to terms with our grief and loss of our beloved son and brother James."
The MoD, which confirmed Cpl Dunsby's death last night, has not revealed where he was from. An inquest into his death is expected to be opened in due course.
Witnesses on the day have described seeing two soldiers "clearly in distress" who pleaded with them for some drinking water.
An inquest at Brecon Law Courts in Powys, mid Wales, gave the medical deaths of L/Cpl Roberts and Mr Maher as "unascertained" and said further tests would be carried out.
L/Cpl Roberts had been pronounced dead on the mountain at 5.15pm, while Mr Maher died at Prince Charles Hospital in Merthyr Tydfil almost three hours later.
Investigations have been launched by both the police and the Health and Safety Executive (HSE).
Powys coroner Louise Hunt will also launch her own investigation, which she said was covered by Article 2 of the Human Rights Act.
She told the court that, under this piece of legislation, the state had "a duty to protect an individual's life".
Ms Hunt also said her investigation was not a "normal" inquest. Any future verdict given must be independent and have the full involvement of both victims' families.
The three men are understood to have been taking part in the aptitude training element of the course to become SAS reservists.
They were climbing Pen Y Fan, the highest mountain in southern Britain.
It is known as the location for the "Fan Dance" where soldiers hoping to join the special forces march over the mountain carrying a heavy pack and a rifle, then do the route in reverse in a set time.
L/Cpl Roberts, 24, of Penrhyn Bay, had served with the Territorial Army for around five years and is understood to have served in Afghanistan and Iraq.
The former teaching assistant lived in London and had been due to start a job in the office of the Education Secretary.
His father Kelvin Roberts said his death left "a massive hole in all our lives", adding: "We wholeheartedly supported Craig in his military endeavours and it gives us some comfort, though great sadness, that he died in the pursuit of his dream."
Mr Maher's family initially asked for him not to be named but issued a tribute ahead of his inquest opening.
They issued a short statement saying: "At this time of great sadness, Edward's family has asked that they be left alone to grieve in private."