How many packages are there in Java SE 7 API?

Oracle includes 209 packages in the Java SE 7 API.

Here is the link to the API specification for the Java™ Platform, Standard Edition:
Java™ Platform, Standard Edition 7 API Specification


Here is the list:

java.applet
java.awt
java.awt.color
java.awt.datatransfer
java.awt.dnd
java.awt.event
java.awt.font
java.awt.geom
java.awt.im
java.awt.im.spi
java.awt.image
java.awt.image.renderable
java.awt.print
java.beans
java.beans.beancontext
java.io
java.lang
java.lang.annotation
java.lang.instrument
java.lang.invoke
java.lang.management
java.lang.ref
java.lang.reflect
java.math
java.net
java.nio
java.nio.channels
java.nio.channels.spi
java.nio.charset
java.nio.charset.spi
java.nio.file
java.nio.file.attribute
java.nio.file.spi
java.rmi
java.rmi.activation
java.rmi.dgc
java.rmi.registry
java.rmi.server
java.security
java.security.acl
java.security.cert
java.security.interfaces
java.security.spec
java.sql
java.text
java.text.spi
java.util
java.util.concurrent
java.util.concurrent.atomic
java.util.concurrent.locks
java.util.jar
java.util.logging
java.util.prefs
java.util.regex
java.util.spi
java.util.zip
javax.accessibility
javax.activation
javax.activity
javax.annotation
javax.annotation.processing
javax.crypto
javax.crypto.interfaces
javax.crypto.spec
javax.imageio
javax.imageio.event
javax.imageio.metadata
javax.imageio.plugins.bmp
javax.imageio.plugins.jpeg
javax.imageio.spi
javax.imageio.stream
javax.jws
javax.jws.soap
javax.lang.model
javax.lang.model.element
javax.lang.model.type
javax.lang.model.util
javax.management
javax.management.loading
javax.management.modelmbean
javax.management.monitor
javax.management.openmbean
javax.management.relation
javax.management.remote
javax.management.remote.rmi
javax.management.timer
javax.naming
javax.naming.directory
javax.naming.event
javax.naming.ldap
javax.naming.spi
javax.net
javax.net.ssl
javax.print
javax.print.attribute
javax.print.attribute.standard
javax.print.event
javax.rmi
javax.rmi.CORBA
javax.rmi.ssl
javax.script
javax.security.auth
javax.security.auth.callback
javax.security.auth.kerberos
javax.security.auth.login
javax.security.auth.spi
javax.security.auth.x500
javax.security.cert
javax.security.sasl
javax.sound.midi
javax.sound.midi.spi
javax.sound.sampled
javax.sound.sampled.spi
javax.sql
javax.sql.rowset
javax.sql.rowset.serial
javax.sql.rowset.spi
javax.swing
javax.swing.border
javax.swing.colorchooser
javax.swing.event
javax.swing.filechooser
javax.swing.plaf
javax.swing.plaf.basic
javax.swing.plaf.metal
javax.swing.plaf.multi
javax.swing.plaf.nimbus
javax.swing.plaf.synth
javax.swing.table
javax.swing.text
javax.swing.text.html
javax.swing.text.html.parser
javax.swing.text.rtf
javax.swing.tree
javax.swing.undo
javax.tools
javax.transaction
javax.transaction.xa
javax.xml
javax.xml.bind
javax.xml.bind.annotation
javax.xml.bind.annotation.adapters
javax.xml.bind.attachment
javax.xml.bind.helpers
javax.xml.bind.util
javax.xml.crypto
javax.xml.crypto.dom
javax.xml.crypto.dsig
javax.xml.crypto.dsig.dom
javax.xml.crypto.dsig.keyinfo
javax.xml.crypto.dsig.spec
javax.xml.datatype
javax.xml.namespace
javax.xml.parsers
javax.xml.soap
javax.xml.stream
javax.xml.stream.events
javax.xml.stream.util
javax.xml.transform
javax.xml.transform.dom
javax.xml.transform.sax
javax.xml.transform.stax
javax.xml.transform.stream
javax.xml.validation
javax.xml.ws
javax.xml.ws.handler
javax.xml.ws.handler.soap
javax.xml.ws.http
javax.xml.ws.soap
javax.xml.ws.spi
javax.xml.ws.spi.http
javax.xml.ws.wsaddressing
javax.xml.xpath
org.ietf.jgss
org.omg.CORBA
org.omg.CORBA_2_3
org.omg.CORBA_2_3.portable
org.omg.CORBA.DynAnyPackage
org.omg.CORBA.ORBPackage
org.omg.CORBA.portable
org.omg.CORBA.TypeCodePackage
org.omg.CosNaming
org.omg.CosNaming.NamingContextExtPackage
org.omg.CosNaming.NamingContextPackage
org.omg.Dynamic
org.omg.DynamicAny
org.omg.DynamicAny.DynAnyFactoryPackage
org.omg.DynamicAny.DynAnyPackage
org.omg.IOP
org.omg.IOP.CodecFactoryPackage
org.omg.IOP.CodecPackage
org.omg.Messaging
org.omg.PortableInterceptor
org.omg.PortableInterceptor.ORBInitInfoPackage
org.omg.PortableServer
org.omg.PortableServer.CurrentPackage
org.omg.PortableServer.POAManagerPackage
org.omg.PortableServer.POAPackage
org.omg.PortableServer.portable
org.omg.PortableServer.ServantLocatorPackage
org.omg.SendingContext
org.omg.stub.java.rmi
org.w3c.dom
org.w3c.dom.bootstrap
org.w3c.dom.events
org.w3c.dom.ls
org.xml.sax
org.xml.sax.ext
org.xml.sax.helpers