I limit this Top 5 to the places I've visited. I couldn't make it to Mrauk U because of unrest at the time I was there, but it certainly belongs to the places to visit as well. Here is more information from Wikitravel.

Another remark is concerning the off limits places: For many places in Burma you need special permits or have to take a tour and that quickly amounts to several 1000 dollars if you really want to see everything. And I wasn't willing to do that, as first it is expensive and second isn't independent travel.

1 Bagan

Why: The main reason is the unique atmosphere

Description: Besides the old temples the thing that attracted me the most here is the atmosphere. The horse carts going their ways, witnessing a temple at sunrise and exploring everything by bike on the second day. It was unique and peaceful. Read more here.

How to get there: Fly in or from Yangon or Mandalay, take the bus from Mandalay or – if you visited Pyay beforehand – come by train from the Shwethekar station 10 km outside of Pyay.

2 Mingun

Why: To see the biggest unfinished pagoda of the world

Description: I was impressed by the size of “the biggest brick-pile of the world”. It looks magic and the earthquake damages add some mystery to it. Also impressive and beautiful is the Hsinbyume Paya. Read more here.

How to get there: Rent a motorbike guide for 15'000 Kyat a day. He will bring you to the place to board the tourist boat across the Ayarwaddy river and will show you other attractions in the afternoon.

3 Hsipaw

Why: I enjoyed hiking through the rice fields.

Description: You should do a hike in Hsipaw through rice fields, local villages and just soak up the atmosphere. Little Bagan in Hsipaw is a plus as you get a chance to see stupas that are not fully restored and hence have their charms. Read more here.

How to get there: You can reach Hsipaw by air by flying to Lashio and then taking a taxi, pick-up or bus to Hsipaw. From Mandalay you can reach it by train in a journey of 10 to 15 hours. By bus from Mandalay you can reach Hsipaw in about 6 hours.

4 Samkar / Inle Lake

Why: The beautiful landscape of Samkar impressed me the most.

Description: After touring the Inle Lake on the first day you can go to Samkar lake south of Inle lake to explore ancient pagodas of the Shan empire. The boat ride down there actually is the main attraction witnessing beautiful landscapes and rural life along the river. Read more here.

How to get there: Tell your guest-house or travel office, that you want to visit Samkar and share a boat with other tourists.

5 Mandalay

Why: It is or feels like the religious centre of Burma.

Description: And because of that offers some of the nicest Pagodas to have a look at, e.g. the Kuthodaw Paya storing the complete Buddhist writings on stone plates. Also don't miss the sunset at Mandalay hill and don't cheat! Going up the stairs for 30 to 45 minutes belongs to the experience. Read more here.

How to get there: Rent a motorbike guide for 15'000 Kyat a day to see the attractions of Mandalay.

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