A lot of the mechanics I ignore...
XP I disregard, inf I treat as money, prestige I disregard...
Levelling I also ignore to an extent, though a new power choice I try to show as Ellie's abilties growing stronger over time.
Enhancements I also disregard for RP, but inspirations I treat as 'pick me up' type pills (so all heroes are junkies!!). Ellie's even accidentally mixed some together after getting KO'd by a rather large CoT behemoth, mixing a large CAB, a large Respite and some Glittershrooms to make what she called a 'Turbo'! (Cue large awaken, large CAB, large Respite, large Rage and a large Breakfree, add buildup and essence boost to that - and a splattered purple behemoth boss!! 
)
Exemping and SKing I see as just teaming with your friends, no increase or decrease of powers.
Respecs... I tend to see them as I've still got the powers I dropped (RP wise), and the new ones just a general increase of ability.
I tend to ignore game time for the most part too, and use realtime, but a lot of RPers do that, and of course - roleplaying is all about fabrication!!
Some of the game backstory I use, some I bend, and some I smash to smithereens!!
@FloatingFatMan
Do not go gentle into that good night.
Rage, rage against the dying of the light.
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How do you explain XP, inf, prestige?
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Xp is entirely OOC.
Inf and Prestige represent the amount of 'pull', influence, or favours either you, or your SG has.
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What are your views on the hero licence and the asociated Security Level?
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The hero license is an IC legal document.
Security Level is an IC concept, an assessment of how capable you are and so what level of threat the city thinks you capable of handling.
There are plenty of ways around needing a specfic SL to get into zones, we bashed this all out ages ago.
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Shops and enhancements, how do you make them work?
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I play a lot of magic heroes so enhancements tend to be little magical gizmos that can empower my spells. They have a reasonable IC justification for all of the basic types so I don't see an issue with them. There's even a little piece about 'fake relics' in the latest CoH comic.
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Exemping and SK?
Training up/leveling up?
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Don't see the issue.
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Respecs?
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Like I said, magic heroes. I tend to just have them mean he's decided to use some different spells.
That said, I've never got a respec from a Trial, only ever the free ones. Trying to justify a heavy dose of radiation having an effect on a magician that doesn't involve a massive cancer infestation isn't easy.
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What do you ignore, what do you fabricate?
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Far too much to list here.
Disclaimer: The above may be humerous, or at least may be an attempt at humour. Try reading it that way.
Posts are OOC unless noted to be IC, or in an IC thread.
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How do YOU make the game work for you, in RP?
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I separate between "game mechanics reality" and "IC reality." In the game mechanics reality Gabbleduck goes through hundreds of minions in a long evening; IC reality he might have deafeated against a handful. In the GM reality he's got Focused Senses, Lucky, Swift, Combat Jumping etcetera. IC reality he's just supernaturally quick, has protective plates of horn on his body that deflects some hits and has got very, very good footwork and technique and mostly fight as a boxer (although as a boxer with claws.) GM reality he has gotten new powers, more hit-points, does more damage, has a higher level etc. IC reality he's gotten better, but only the kind of limited improvement that comes from honing one's technique and getting more experience. GM reality he does the same mission archs and fights the same elite bosses and AVs as everybody else. IC reality he fights against "private" bosses and do "private" missions -- the GM missions are just representative or symbolic of the IC missions.
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How do you explain XP, inf, prestige?
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I don't. Inf, sometimes, I hand-wave away as "favours owed."
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What are your views on the hero licence and the asociated Security Level?
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There is an IC bueraucracy complete with licenses, security levels and similar; but there are those who can work around it (such as Nevermore.) They got their own ways around the security checks between the zones and similar.
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Shops and enhancements, how do you make them work? (if at all)
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Don't exist. Gabbleduck's improvement doesn't come from enhancements, but from training and experience.
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Exemping and SK?
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Batman and Robin.
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Training up/leveling up?
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Doesn't exist. Gabbleduck has probably never spoken to either Backstreet Brawler or any of the other "government heroes."
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Respecs?
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Changing his fighting style to improve it.
I have no problem using most of the above as IC, as they made a good try at giving them nice IC justifications when they made this game.
XP I ignore its purely a games mechanism.
Prestige and influence represent what they say 
as heroes we are ever in the public eye..for good or bad.
I love the hero licence,reminds me of an episode of "The Tick" live action series..hehehehe.
Enchancments in the most cases are items or just skill development..
One thing Id love to see in the comics is warwitch with all her enhancments on, as a magicer she has to wear ear rings,bracelts,bands,candles,rings.... No stealthy for her methinks.
exemping and sking just represent in once case "the passing on of knowledge and tactics to a young apprentice,thus improving their skill" and in the other case...hmmm I'll get back with that one...
Respecs... they dont work that will IC, some ATS anc character concepts can justify them well (swapping equipment,learning new spells,mutating further and losing an old ability to gain a new) but others cannot at all.
So I tend to ignore them.
XP - Understanding your powers.
Influence - Fame
Prestige - Media Fame
Hero Licence : ID Licence
Security Level : Age in Paragon fighting
Enhancements - Training
("Whilst studying an arcane ritual, I found a process that would allow me to conduit the mana directly into the body without bypassing the air")
Exemping : "Don't worry kid, I'll show you how"
SK : "Gee wow...can I do that?"
Training up/levelling (Happens across the level..I develop)
Respecs : Hate respecs.
I "rationalize" xp as a measure of your "reputation" as a villain. Once your reputation goes high enough, you are deemed to to higher ranked security/threat level. Debt comes easy from this, as failures aren't good for reputation. I ignore combat level. Haven't really given that much thought to influence or prestige, since I hardly ever need to talk about them.
I try to keep game mechanics as seperate from my IC roleplaying as possible. As others have said, things like Hero Licences and Security Levels have an IC presence in the game, but I don't RP that my character pops down to the local Training Shop in Talos to attach an Enhancement to an ability.
I usually don't even take a character's actual level into account when considering how powerful they are. If the player is a good roleplayer and someone I trust, I'll let them act as powerful or as weak as they want!
How do YOU make the game work for you, in RP?
How do you explain XP, inf, prestige?
What are your views on the hero licence and the asociated Security Level?
Shops and enhancements, how do you make them work? (if at all)
Exemping and SK?
Training up/leveling up?
Respecs?
What do you ignore, what do you fabricate?
@ShadowGhost & @Ghostie
The Grav Mistress, Mistress of Gravity
If you have nothing useful to say, you have two choices: Say something useless or stay quiet.