Determinator/Music

"500 Miles" by the Proclaimers
And its Spanish similar of sorts, Dúo Dinámico's "Resistiré".
"Eye of the Tiger" by Survivor
"It's My Life" by Bon Jovi
- Also by Bon Jovi, "Livin' On a Prayer."
"Stay Hungry" by Twisted Sister
Pick a song by DragonForce. Any song.
"No Surrender", and "Unstoppable" (among others) by Bone Thugs-n-Harmony.
"Stand" by Motorhead
"Indestructible" by Disturbed.
Queen's "The Show Must Go On" not only fits this trope in lyrics, but also in purpose; it was intended as Freddie Mercury's last defiant stab at death. It was released shortly before he died of AIDS.
Pat Benatar's Invincible.
"Until the End" by Breaking Benjamin. Also, "I Will Not Bow."
"Curse of Feanor" by Blind Guardian - it is about Fëanor, after all. You can find his entry in the literature section.
Stand by Rascal Flatts.
Like a Rock by Bob Seager.
- Charging from the gate?
Most anything by Hammerfall - especially "Hearts on Fire" and "Heeding the Call".
"I Won't Back Down" by Tom Petty

You can stand me up at the gates of hell
But I won't back down

Any metal drummer. Pick one. Their parts are usually incredibly active and some play at blistering speeds, even given the genre.
"Spirit Never Dies" by Masterplan:

"Never give up/ Never give in/ Won't stop believing/ 'Cause I'm gonna win [...] There is no limit to what can be done! / Climbing the mountain with power so strong"

"Tubthumping" by Chumbawumba
Metal band Anvil -- this is a band that has been active for over thirty years, but had probably the worst luck you'll ever see, but did this stop them? No! They kept pressing on until they finally got a break in 2008 when their documentary was able to help bring them the success they deserve.
- At least they got their documentary, although mainstream media downplayed it as an example of Giftedly Bad. Riot have had even worse luck: they had to sue their record label to get an album released, and then were dropped. Fire Down Under was delayed for a year by lawsuits: it wasn't exactly a hit, but is often mentioned as the first Speed Metal album, and is to this day considered a classic. Then singer Guy Speranza quit music entirely. After recording one album with Rhett Forrester on vocals, they got dropped by their record label again. It was discovered later that the record labels weren't even their biggest problem: their own management team acted in a way that scared off anybody who wanted to give Riot higher-level promotion, because their management was more interested in having a "pet" band than having a successful one. Among the promoters chased off were Aerosmith's management team, who were looking to expand their roster at a time that Aerosmith were semi-disbanded. They eventually found another label, but in the meantime, the similarly-named Los Angeles band Quiet Riot had gone platinum, causing an identity problem for the New York City-based Riot. Riot broke up, and resurfaced in Texas several years later with only guitarist Mark Reale remaining from their "classic" lineup. They continue to this day to be a well-respected band, but are mainly able to keep going because they are Big in Japan. Also, continuing the blues refrain, no hopes of reunions of their early lineups, as Speranza and Forrester have both since passed.
mc Chris's "Never Give Up."
You only realize this if you listen to the lyrics but Welcome To The Black Parade by My Chemical Romance. The chorus is basically "We'll Carry On" over and over again to insane levels, with lines about not giving up and stuff of that type.
Emilie Autumn lives this trope
Britney Spears shows element of this trope.
Eminem's "Lose Yourself".
Martine McBride's "Concrete Angel"

"Through the wind and the rain she stands hard as a stone
In a world that she can't rise above"

"The Distance" by CAKE. The guy's already lost the race. He's just trying to finish.
Styx's "Blue Collar Man (Long Nights)" is about an desperate unemployed man who is willing to work as long and as hard as he can to hold a steady job.
I Will Survive by Stephanie Bentley
"Vandraren" (The Wanderer) by Nordman.
Don't Give Up by Eagle Eye Cherry