Since making his Test debut in 2000, Graeme has survived a turbulent last four years intact - or at least patched back together - and still harbours a burning desire to prove he is the best scrum-half in Scotland.
The campaign for a Test place, which began with a solid display in the second half of last Saturday's defeat to Queensland, will continue in earnest on Wednesday night when Scotland take on NSW Country.
"I play rugby because I want to play for my country, so any time you are selected to go on a tour or in a squad, that is the biggest thrill," he said.
"I was on the bench against the Barbarians (on May 22) and running out at Murrayfield again for the first time in a while was just brilliant. You never get sick of that and it makes all the injuries and other stuff worthwhile.
"There are two very good young scrum-halves coming through now but I am still only 28 and I certainly don't think I am coming to the end. With two guys in front of me there is obviously pressure every game. Every time you go on the park you have got a point to prove and an opportunity to prove you are good enough to play in the Test team."
The competition is fierce with Chris Cusiter and Mike Blair ahead of him, but then it always has been.
Things might well have been different had Graeme not spent a total of 20 months out recovering from successive dislocated ankles.
After winning his first cap off the bench against New Zealand in 2000 it took three years for Graeme to start his first Test for Scotland, in last August's World Cup warm-up against Wales.
"The first time I was injured I was down for a long time during my rehab but I read a very good article on the psychology of injury and it was interesting how it can almost destroy you if you let it get on top of you," Graeme recalled.
"You can start to worry about things you can't control, about whether you are playing better than anyone else. You lose focus on what you are trying to do. Ultimately it is how you perform as an individual, to play the best you can and get selected, not how other guys are performing."
With the team for the first Test against Samoa named before kick-off on Wednesday, a stand-out display against NSW Country is unlikely to earn Graeme an immediate look-in. But there are two Tests against Australia to come and Scotland head coach Matt Williams indicated to the players that places in every position will go to the man in form.
"Everybody who is coming on this tour has a chance to play for Scotland," said Beveridge. I am just going to try and play the best I can and put my hat in the ring for the Test team, whether it is for this game against Samoa or the Test against Australia."
After Scotland suffered their eighth defeat on the bounce to Queensland, victory on Wednesday is now vital to the squad's momentum and morale heading into Friday's Test.
"Wednesday is a huge game for the whole tour because we are desperate to get a win under our belts so we can build from it as a squad," said Graeme.
"It is amazing what one win can do and if we get that win on Wednesday and take that into the Test on Friday then who knows what will happen?
"It is important to get a win and we can start building momentum as a squad."