Bishan Campground (aka Bishan Camping Area) in Neihu District of Taipei City is the only free campground run by Taipei City. Near the campground is Bishan Temple, which has one of the best views of Taipei. Besides camping and temple worship, the area is also a popular hiking spot, and there are multiple tourist farms and tea shops to explore. 

How to apply for a camping spot:
You can apply using this online form (it's in English!):
 http://gisweb.taipei.gov.tw/TPCamp/en/ABishan.aspx
You must apply at least 7 days in advance, but no more than 30 days in advance. 
You must enter your passport ID or Taiwan ID. 
You can check the status of the application online as well. 
For more info, check here (Chinese): https://www.geo.gov.taipei/mp10504b/ 

Price:
Free 

Utilities:

• Free showers
• Free electricity (one or two outlets)
• Free parking
• Bring your own tent, sleeping gear, and food. 

Hours:
Camping time starts at 2 PM and ends at 1 PM the next day. 
You can only stay for two nights at a time. 

How to get there:
By Bus: You can take a bus to about 1 KM away from the campsite, but then you have to walk with all of that gear. If that's not what you want then you should take a taxi or your own vehicle. 
By Scooter/Car: From East Neihu, drive up Bishan Raod (碧山路) until you see the campground. It's hard to miss. 

Map:
Please see below:

Keelung's Miaokou Night Market is a large night market in northern Taiwan, full of all kinds of delicious Taiwanese street food considered by some to be the best in Taiwan. The market consists mostly of food stalls in front of the Dianji Temple  in Keelung, streching on both sides down the street. It is said to be a must see night market and one of the busiest in Taiwan. 

Background:
The Miaokou Night Market started up after completion of the Dianji Temple (奠基宮) during the Qing Dynasty in 1873. At the time the temple was surrounded by rice fields, but by the Japanese Era the area around the temple began to be developed. 
Food stalls really started to spread out at the entrance of the temple at the end of the Japanese Era. Since then, the famed food at Miaokou has brought more and more tourists to the area seeking delicious snacks.
In recent years (since 2017), many stalls at the night market have become vacant due to high rents, and there have been notably less people visiting the night market. Some say that this is because of the advent of online shopping and food delivery apps. However another big factor is overpriced properties and stingy landlords. The night market is not going to close down anytime soon, but Taiwan is going through an economic transition period that is not only affecting the rental market but also consumer trends. Despite this, you will still be able to find delicious food on the street any night of the year. 

Hours:
Every day 12:00 noon to 12:00 midnight. 

Price:
Free admission.

How to get there:
By Train: Take the TRA to Keelung Station, then walk east on Zhongyi Road (忠一路) until you reach the night market. 
By Bus: Taking Keelung Bus (intercity bus) or K-Bus from Taipei Main Station or City Hall to Keelung. 
By Car: Take National Highway 1 north to the end of the highway until it becomes central Keelung, then turn right. There is plenty of parking to the south of the night market underground. 
By Scooter: Take provincial highway 5 to central Keelung. There is parking along the street but you might have to walk a ways to be able to park in a legal spot. 
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Map:
Please see below: