The Face of Boe is considered an "immortal" but he does eventually die. In the Doctor Who universe, books, TV, and comics, was an exact age of the Face of Boe given when he died in "Gridlock"?
Five billion, give or take a few significant figures.
Pure on-screen canon
We can't nail down an exact age, but since several of his appearances are dated we can come up with a roundabout figure. In (in-universe) chronological order:
"The Long Game" is said to take place in the year 200,000, according to pretty much the first line of dialogue. The Face of Boe appears momentarily on a news screen, and is apparently pregnant:
Cathica: Latest news, sandstorms on the new Venus archipelago. Two hundred dead. Glasgow water riots into their third day. Space lane seventy seven closed by sunspot activity. And over on the Bad Wolf channel, the Face of Boe has just announced he's pregnant.
"The End of the World" is of course our first exposure to Boe, and the episode is stated to have taken place "five billion years in [Rose Tyler's] future. What's not entirely clear whether that's the year 5,000,000,000 or 5,000,002,005. Either way, long time in the future.
"Gridlock" is explicitly said to take place in the year five billion and fifty-three, which retroactively narrows the time period when "The End of the World" could have happened:
Doctor: I don't want to go home. Instead, this is much better. Year five billion and fifty-three, planet New Earth. Second hope of mankind. Fifty thousand light years from your old world, and we're slap bang in the middle of New New York. Although, technically it's the fifteenth New York from the original, so it's New New New New New New New New New New New New New New York. One of the most dazzling cities ever built.
Based purely on what is confirmed on-screen, the Face of Boe is at least 4,999,800,053 years old. However, since in "Bad Wolf" (which takes place in the year 200,100) he was said to be the "oldest inhabitant of the Isop galaxy", he's likely much older than this.
This meshes with a comment made by the Doctor in "Girdlock" regarding Boe's supposed longevity:
Doctor: Legend says the Face of Boe has lived for billions of years. Isn't that right? And you're not about to give up now.
Using some meta-textual1 evidence
If we accept the scads of interview evidence saying that the Face of Boe is Jack Harkness, we can do a bit better. Although we're not sure exactly how old Jack is when we first meet him on the show2, we know that he eventually wound up in 1869 without a time machine:
Jack: All right, so I bounced. I thought 21st century, the best place to find the Doctor, except that I got it a little wrong. Arrived in 1869, this thing burnt out, so it was useless.
Of course he gets his Vortex Manipulator fixed by the Doctor about a century-and-a-half-later3, but by that point he's already around 180 years old. There's one time-travel related complication here, from the Torchwood episode "Exit Wounds"4.
In that episode (which takes place in about 2008), Jack is sent back in time to the year 27, where he's buried alive until 1901, at which point he's placed in suspended animation until 2008 again.
I'm not going to count the century he was in suspended animation, because he wasn't really ageing at the time, but if you want to then just remove 107 from whatever number I come up with.
Going off my assumption that he was about 30 when he got sent back to 1869 (which may be a stretch, but probably not much of one), he'd be 5,000,000,028 years old when he died.
1 That's me putting on my lit-crit pants
2 In an episode of Torchwood it's revealed that he was born around the year 5094, but we don't know when he went back to World War 2 London
3 Although he loses it at some point, since it appears in the Black Archive in 2010 during the events of "The Day of the Doctor" and River Song is implied to have stolen it from Dorium in "The Pandorica Opens"
4 Thanks to Richard for pointing this one out in comments
We can't be sure the Face of Boe didn't do any time-traveling for himself, but in Gridlock the Doctor does say "Legend says the Face of Boe has lived for billions of years." As for the interview evidence that Jack is the Face of Boe, please note showrunner Russell T Davies' comments here that he originally put in Jack's line because "it made me laugh", that it was deliberately "couched in terms that are not absolute gospel", and that he rejected any spinoff ideas that would have made it more definitive. Jun 7, 2015 at 2:22
@Hypnosifl Good point regarding the Doctor's line in "Gridlock"; I'll add that in in a moment. Regarding Davies' comments: you're right, but other people connected with the show have come out and said that Jack was Boe: John Barrowman, Julie Gardner, and David Tennant have all done so. Regardless it's not unambiguously clear, hence why it got its own section Jun 7, 2015 at 2:25
1Wasn't jack buried for a few hundred years as well?– ValorumJun 7, 2015 at 15:25
@Richard He was indeed; missed that one on my skim. Added it in, and now we're tantalizingly close to a round "5 billion" Jun 7, 2015 at 15:42
@JasonBaker - Didn't he also spend all eternity stuck on the outside of the TARDIS once?– ValorumJun 7, 2015 at 15:45