It’s no secret Copenhagen can be hard on the pockets but, then again, it’s also known as one of the world’s most liveable cities…

You’ll be surprised how much there is to do in the Danish capital without having to spend two bits. Here are our favourites:

Kick back in Christiania

Did you know that, technically, you can leave Copenhagen without actually leaving? We’ll explain.

In 1971, a group of anti-establishmentarian hippies occupied an area of abandoned buildings in the centre of the city and established the self-governed Freetown Christiania.

Though this small patch of land does generate some controversy – and you should take care to respect local rules, such as restrictions on photography – Christiania is about much more than pot-shopping on Pusher Street.

A cherished refuge for locals, the landscape is beautiful, the atmosphere is laid-back and there are art galleries, music venues and cheap, organic eateries scattered throughout.

Walking the dusty paths among the trees spotting wacky local houses or sitting by the lake will, of course, cost you nothing and, in the summer, the volunteer-run Café Nemoland puts on free concerts every Sunday, showcasing everything from Danish reggae and dancehall to playful re-interpretations of old protest songs by some of the city’s most famous artists.

Take a breather in the Botanical Garden

Copenhagen’s Botanical Garden contains the largest collection of living plants in the country and serves both as a research area for the city’s zealous botany students and as a green oasis for summer picnics and lying peacefully in the grass.

If the sun isn’t out, you can warm up in the tropical climate of the “palm house”, one of four large greenhouses on the property, housing rare plants like the Indonesian Amorphophallus Titanum. Can you guess from the name what it resembles?

Beware this exotic fiend when it’s in bloom though; the stench it exudes has earned the giant plant its other, equally unflattering, moniker: “corpse flower”. Yuck.

Jump head first into the harbour

Every summer since it first opened, the Harbour Bath at Islands Brygge has been a favourite spot for hordes of overheated Copenhageners who want to cool down and relax after a hard day’s work.

Recently, the open air baths have expanded to include a winter pool, so those with a Viking temperament can enjoy a plunge all year round.

Even if dipping your toes into chilly waters isn’t your idea of fun, the stunning city skyline seen from the waterfront is still worth the trip.

Sleek modern buildings reflect the sunlight, boats bob along the silver water, the twisting verdigrised spires of the Old Stock Exchange jut out into the sky and, last but not least, The Black Diamond – the Royal Danish Library – shines bright and dramatic across the bank.

Explore the grungy Nørrebro neighbourhood

For a look underneath the Copenhagen’s pretty, cobblestoned surface, mount your bike and head to Nørrebro. Gritty, hip, casual and the most culturally diverse neighbourhood in the city – it’s people-watching-perfect.

Have a picnic between H.C. Andersen and Søren Kierkegaard in the cherished Assistens Cemetery. Browse the weekend flea market set up along its bright yellow outer wall, or window-shop on nearby eco-and-design-obsessed Jægersborggade.

Stop by the polychrome city space Superkilen for a spontaneous boxing match. Watch the cool kids skating in Nørrebroparken. Pet a goat at the ByOasen urban garden. Listen to boom boxes blaring music on Dronning Louise Bridge with the locals.

You’ll feel spoilt for choice.

Walk the city streets

Cycling is indisputably the best way to get around the Danish capital but the immensely popular Copenhagen Free Walking Tours are also a great way to explore the city.

Tips aren’t mandatory but you’ll want to leave a little something to the enthusiastic team of locals winding you through the city’s streets and stories, determined to pass their passion for Copenhagen onto you.

Choose between a classic, exhaustive 3 hour tour that leaves no Copenhagen (cobble)stone unturned, or two 90-minute expeditions – one focused on the city’s historical landmarks, the other centered around the cosy Christianshavn area and neighbouring Christiania.

Skip town for a day

If, on the other hand, the biking culture has charmed you and you want to take it up a notch, the 45 minute cycle to the quaint white-and-blue themed Bellevue Beach makes for a beautiful scenic ride – and your destination is a masterpiece of modernist architecture. You’ll have to spend a little on renting a bike but there are some reasonably priced options, including Baisikeli, whose proceeds go toward making bicycles accessible in African countries.

From the functionalist petrol station to the iconic theatre with its convertible roof, the small changing rooms on the sand and the soaring lifeguard towers along the seaside, virtually all the buildings of this “white town” were created by world-famous Danish architect Arne Jacobsen, after winning a competition to design the resort in the 1930s.

When you’re done admiring the coastline, the 1000 hectares of scenic, cultivated woodland in Dyrehaven (The Deer Park) await you just a stone’s throw from the beach.

Along with shy fawn hiding in the scrubland, it’s also home to the world’s oldest operating amusement park, Bakken – it’s like Tivoli, only free to get into.

Looking for more inspirational and FREE ways to explore a city? Check out the momondo blog.

Want more info about the great things you can do in Copenhagen? Download our FREE city guide for your iPhone via the App Store.