This list will date quickly as new books are published (last updated December, 2011) - many titles have been abbreviated
Most of these books are probably necessary, but I have flagged a couple as definitive
BI foundations - Star Schema
Star Schema - Christopher Adamson - McGraw-Hill (definitive)
Microsoft BI
Delivering Business Intelligence - Brian Larson - McGraw-Hill
Microsoft Data Warehouse Toolkit - Joy Mundy et al - Wiley
Microsoft SQL Server 2008 BI 70-448 - Erik Veerman et al - Microsoft
SSAS cubes
Applied Microsoft Analysis Services 2005 - Teo Lachev - Prologika
Microsoft SQL Server 2008 Analysis Services - Irina Gorbach et al - Sams (definitive)
Expert Cube Development - Chris Webb et al - Packt
Microsoft SQL Server 2008 Analysis Services - Scott Cameron et al - Microsoft
MDX
Microsoft SQL Server 2008 MDX - Bryan Smith et al - Microsoft
MDX Solutions - George Spofford et al - Wiley
MDX with R2 SSAS - Tomislav Piasevoli - Packt
Practical MDX Queries - Art Tennick - McGraw-Hill
SSAS data mining
Data Mining with Microsoft SQL Server 2008 - Jamie MacLennan et al - Wiley (definitive)
DMX
Data Mining with Microsoft SQL Server 2008 - Jamie MacLennan et al - Wiley (definitive)
Practical DMX Queries - Art Tennick - McGraw-Hill
XMLA
Microsoft SQL Server 2008 Analysis Services - Irina Gorbach et al - Sams (definitive)
Practical XMLA Queries - Art Tennick - ? (hopefully, published 2012)
PowerPivot for Excel 2010
Professional Microsoft PowerPivot for Excel and SharePoint - Sivakumar Harinath et al - Wrox
Practical PowerPivot and DAX - Art Tennick - McGraw-Hill
DAX
PowerPivot for Excel 2010 - Marco Russo et al - Microsoft
Practical PowerPivot and DAX - Art Tennick - McGraw-Hill
PowerPivot for SharePoint 2010
Professional Microsoft PowerPivot for Excel and SharePoint - Sivakumar Harinath et al - Wrox
PerformancePoint
Microsoft SharePoint 2010 PPS - Tim Kashani et al - Sams
Monitoring and Analyzing with PPS 2007 - Nick Barclay et al - Rational Press
BI with PPS 2007 - Craig Utley - McGraw-Hill
Pro PPS 2007 - Philo Janus - Apress
SSIS
Professional Microsoft SQL Server 2008 Integration Services - Brian Knight et al - Wrox
SSRS
Microsoft SQL Server 2008 Reporting Services - Brian Larson - McGraw-Hill
Microsoft SQL Server 2008 Reporting Services - Stacia Misner - Microsoft
SQL
Accelerated SQL Server 2008 - Robert Walters et al - Apress (intermediate/advanced)
Practical SQL Queries - Art Tennick - McGraw-Hill (introductory)
Excel
Excel 2010 Inside Out - Mark Dodge et al - Microsoft
Excel 2007 Data Analysis and Business Modeling - Wayne Winston - Microsoft
Office
Microsoft Office 2007 BI - Doug Harts - McGraw-Hill
Statistics (not really essential BI - but useful background for data mining and DMX)
Statistical Techniques in Business and Economics - Douglas Lind et al - McGraw-Hill (uses Excel)
Data Analysis and Decision Making - Christian Albright et al - Thomson (uses Excel)