These are the quest listings that need to be completed in order to raise your Hunter Rank.

HR 1 - HR 2
3*
[Giadrome Assault]
- Hunt the Giadrome

[The Land Shark]
- Hunt the Cephadrome

[The Lurking Desert Giant]
- Hunt the Daimyo Hermitaur

[The Mischief-maker]
- Hunt the Congalala

Urgent Quest
4*
[The Ruler of the Snow]
- Hunt the Blangonga

HR 2 - HR 3
4*

[The Shadow in the Cave]
- Hunt the Khezu

[Master of the Giant Lake]
- Hunt the Plesioth

[Evening Hermitaur Sonata]
- Slay 20 Hermitaurs

[Pincer through the Sky]
- Hunt the Shogun Ceanataur

[Trouble in the Forest]
- Slay 20 Bullfangos

[The Ioprey Leader]
- Hunt the Iodrome

Urgent Quest
5*
[Absolute Power]
-Hunt the Tigrex

HR 3 - HR 4
5*

[The Runaway Diablos]
- Hunt the Diablos

[Valor in the Swamp Zone]
- Hunt the Gravios

[The King's Domain]
Hunt the Rathalos

[The Queen's Descent]
- Hunt the Rathian

Urgent Quest
5*
[The Approaching Gaoren]
- Defend the Fortress from Shen Gaoren

HR 4 - HR 5
6*

[The Poison Fanged Duo]
- Hunt 2 Gendrome

[Ultimate Crab Dinner]
- Hunt 2 Daimyo Hermitaurs

[Trapped by Yian Kut-Ku]
- Hunt the Yian Kut-Ku and Blue Yian Kut-Ku

[Conga Counterattack!]
- Hunt the Congalala

Urgent Quest
7*
[Lao-Shan Lung Draws Near!]
- Defend the Fortress from Lao-Shan Lung
(Ash Lao-Shan Lung)

HR 5 - HR 6
7*

[Two Roars in the Snow]
- Hunt 2 Blangongas

[Red Shadow in the Swampland]
- Hunt the Red Khezu

[The Underwater Terror]
- Hunt the Green Plesioth

[Elminate the Rathalos!]
- Hunt the Rathalos

[Basarios: Unseen Peril]
- Hunt the Basarios

Urgent Quest
8*
[Land of the Tremors]
- Hunt 2 Tigrex

HR 6 - HR 7
8*

[The Fierce Black Horn]
- Hunt the Black Diablos

[Black Rock in the Swamp]
- Hunt the Black Gravios

[Blue Sky, Pink Earth]
- Hunt the Azure Rathalos and the Pink Rathian

[Deny the Silver Rathalos]
- Hunt the Silver Rathalos

[Find the Golden Phantom]
- Hunt the Gold Rathian

Urgent Quest
8*
[The Approaching Gaoren]
Defend the Town from Shen Gaoren

Urgent Quest
8*
[Rise to the Summit]
- Slay the Akantor

Urgent Quest
G*
[Hypno, Hypno!]
- Hunt 2 Hypnocatrices

HR 7 - HR 8
G*

[Daimyo of Sengoku Fame]
- Hunt the Plum D. Hermitaur

[Old Jungle Lightning]
- Hunt the Khezu

[Eyes in the Underground Lake]
- Hunt the Plesioth

[Pelagus Pride]
- Hunt the Emerald Congalala

[Old Swamp, Shrouded in Mist]
- Hunt 2 Purple Gypceros

Urgent Quest
G**
[Ever-Present Shadow]
- Hunt the Nargacuga

HR 8 - HR 9
G**

[Lavasioth Sighting!]
- Hunt the Lavasioth

[Under the Gaze of Heaven]
- Hunt the Shogun Ceanataur and the Terra S.Ceanataur

[A King, Robed in Smoke]
- Hunt the Rathalos

[Her Eternal Majesty]
- Hunt the Rathian

[Wild Monk of the Dunes]
-Hunt the Copper Blangonga

Urgent Quest
G***

[The Floating Disaster]
- Slay the Yama Tsukami


To Unlock Ukanlos
G***

[Absolute Power]
- Hunt the Tigrex

[Black Phalanges]
- Hunt 2 Black Diablos

[Armored Supremacy]
- Hunt the Gravios

[Explore the Unexplored]
- Hunt the Silver Rathalos

[A Single Beam of Moonlight]
- Hunt the Gold Rathian

[The Approaching Gaoren]
- Defend the Town from Shen Gaoren


Urgent Quest
G***
[Absolute Zero]
- Slay the Ukanlos


Source:Wikipedia