I've seen people get free ones on their birthday but it's because the bartender or another patron buys it for them. But by Oregon liquor law, it's not permitted for a bar to give free drinks for a birthday (Most likely because it would promote drinking and driving while bar hopping to get tons of free alcohol.) Remember that even public intoxication is technically illegal, so bars aren't supposed to be meant for getting drunk, but to drink socially and get a buzz but still maintain composure.
It's not permitted by OLCC. But someone it's very likely to buy you a birthday drink if you mention it.
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