The 24-year-old, who has played for Greece at junior level, joins on a free transfer from Olympiacos - also owned by Forest owner Evangelos Marinakis.
Bouchalakis had been expected to sign a loan deal with Forest.
But he impressed in training and in friendlies with Notts County and Mansfield and has signed until 2020.
Find all the latest football transfers on our dedicated page.

According to his spokeswoman, "Matthew may tape something" for the tribute to the show's creator James Burrows.
"In other words, this is not the reunion people have been hoping for," Lisa Kasteler told Us Weekly.
His former co-stars, Jennifer Aniston, Courteney Cox, Lisa Kudrow, David Schwimmer and Matt LeBlanc are all expected to take part in the special.
It is set to be aired on NBC on 21 February.
Friends ended in 2004 after 10 series. As well as Friends, Burrows worked behind the camera on sitcoms including Cheers and Frasier.
Perry has written and will star in the play The End of Longing... which is set to run in London's West End from 2 February until 14 May. It will be directed by Lindsay Posner.
The dark comedy follows four disparate people approaching middle age, who meet in a bar one night.
It marks Perry's first return to the West End since 2003, when he starred in David Mamet's Sexual Perversity in Chicago.

The company reported an 8% increase in profit to $391m (Â£273m) for the first three months of 2016.
Yum, which also owns Pizza Hut and Taco Bell, said sales at stores in China that have been open a year or more rose 6% compared to the same quarter last year.
The news boosted Yum's shares by 4% in after-hours trading.
Analysts had expected just a 2.1% growth in sales in China, where Yum is the biggest Western restaurant brand.
Chief executive Greg Creed said: "KFC China had an outstanding Chinese New Year bucket promotion."
"This is a transformational year for our company as we remain on track to finalise the separation of our China business by year end," Mr Creed said.
Yum wants to spin-off its 6,900 China restaurants, its most profitable business, by the end of 2016 in order to focus the company on US operations.
It has also been losing market share to rival McDonalds in China.

Laws, 26, had been training with Arsenal but has decided to remain in the north east by swapping Sunderland for WSL 2 club Durham.
She will fill in for regular goalkeeper Helen Alderson, who is going to Africa for a three-month charity work stint.
"It's a club going in the right direction," Laws said. "I can't wait to start with a new club."
She continued: "Obviously it is a bit strange for over 10 years but it's a challenge I'm looking forward to getting started with."
The Newcastle-born keeper played 18 games for the Lady Black Cats last season, where she has played the majority of her career barring a loan spell at Liverpool in 2013.
Nottingham Forest have signed Andreas Bouchalakis on a three-year contract after the midfielder impressed during pre-season with the Championship side.
[NEXT_CONCEPT]
Matthew Perry "will not" attend the Friends reunion show due to rehearsals for his West End play.
[NEXT_CONCEPT]
KFC owner, Yum Brands, has seen a rise in profit, thanks in part to a chicken bucket deal over Chinese New Year.
[NEXT_CONCEPT]
Durham Women have signed England international goalkeeper Rachel Laws for the WSL Spring Series competition.