Sorry, I forgot to mention that I am using TB 38.0.1 on Mac OS X 10.10.4 on two different Macs, both exhibit the same issue.

Try this add-on. https://addons.mozilla.org/en-US/thunderbird/addon/startupmaster/

Jebus, so bug #177175 seems to have arisen. Weird, I've never had the issue with any release of Firefox, nor any version of Thunderbird prior to 38.0.1 (and I've been running Firefox and Thunderbird for many years now). I've also confirmed that the issue affects TB 38.0.1 on Ubuntu 15.04.

I'll try the add-on, thanks for pointing me to it, but it seems crazy that it's necessary.

I don't disagree. In any case, if this did fix your problem can you mark the thread as 'Solved' please? Thank you.

Sigh. StartupMaster does fix the multiple-master-password-entry problem, however it corrupts the Thunderbird application menu - with StartupMaster activated when you click on the Thunderbird menu the only option displayed is "Quit Thunderbird", no "About Thunderbird", "Preferences", "Services", etc. entries are shown. Disable StartupMaster and all is well with the Thunderbird menu again. I've emailed the add-on's author about this issue.

So no, it's not resolved.

I do use StartupMaster myself, and don't see such a problem. Are you using the default theme? Can you post your Troubleshooting Information? Help (Alt-H) - Troubleshooting Information

Yes, I'm using the default theme. Troubleshooting info below:

 Application Basics

   Name: Thunderbird
   Version: 38.0.1
   User Agent: Mozilla/5.0 (Macintosh; Intel Mac OS X 10.10; rv:38.0) Gecko/20100101 Thunderbird/38.0.1
   Profile Folder: Show in Finder
   Application Build ID: 20150608103712
   Enabled Plugins: about:plugins
   Build Configuration: about:buildconfig
   Memory Use: about:memory

 Mail and News Accounts
   account2:
     INCOMING: account2, , (none) Local Folders, plain, passwordCleartext

   account6:
     INCOMING: account6, , (imap) imap.googlemail.com:993, SSL, OAuth2
     OUTGOING: smtp.googlemail.com:465, SSL, OAuth2, true

   account8:
     INCOMING: account8, , (imap) imap.mail.me.com:993, SSL, passwordCleartext
     OUTGOING: smtp.mail.me.com:587, alwaysSTARTTLS, passwordCleartext, true

 Crash Reports

 Extensions
   Enigmail, 1.8.2, true, {847b3a00-7ab1-11d4-8f02-006008948af5}
   ImportExportTools, 3.2.1, true, {3ed8cc52-86fc-4613-9026-c1ef969da4c3}
   LookOut, 1.2.13, true, lookout@aron.rubin
   Show All Body Parts, 1.2, true, ShowAllBodyParts@kamens.us
   StartupMaster, 1.6.1-signed, true, {506d044e-41fa-4cc8-9dc6-9ff70e96eebf}

 Important Modified Preferences

   Name: Value

     accessibility.typeaheadfind.flashBar: 0
     browser.cache.check_doc_frequency: 1
     browser.cache.directory: AAAAAAEUAAIAAQRFbGxlAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACrp4FYQkQAAAAEqDIFQ2FjaGUAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA…
     browser.cache.disk_cache_size: 10000
     browser.cache.disk.capacity: 9216
     browser.cache.disk.parent_directory: AAAAAAGWAAIAAQdDbGF1ZGlhAAAAAAAAAAAAAAAAAAAAAAAAAAC3RzyKSCsAAAACdFQMaHl3NXA2eHEuc2x0AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA…
     browser.cache.disk.smart_size.enabled: false
     browser.cache.disk.smart_size.first_run: false
     browser.cache.disk.smart_size.use_old_max: false
     browser.cache.memory_cache_size: 2048
     browser.cache.memory.capacity: 30720
     browser.display.screen_resolution: 96
     browser.display.use_document_fonts: 0
     browser.startup.homepage: http://www.bigpond.com
     browser.startup.homepage_override.1: false
     browser.startup.homepage_override.mstone: rv:1.7
     browser.tabs.opentabfor.windowopen: true
     browser.tabs.warnOnClose: false
     browser.zoom.full: true
     dom.disable_open_during_load: true
     dom.disable_window_flip: true
     dom.disable_window_move_resize: true
     dom.event.contextmenu.enabled: false
     dom.max_chrome_script_run_time: 300
     extensions.lastAppVersion: 38.0.1
     font.internaluseonly.changed: false
     font.minimum-size.x-western: 10
     font.name.sans-serif.x-western: Verdana
     font.name.serif.x-western: Times New Roman
     font.size.variable.x-western: 14
     gfx.blacklist.direct2d: 3
     gfx.blacklist.layers.direct3d9: 3
     mail.openMessageBehavior.version: 1
     mail.spotlight.firstRunDone: true
     mailnews.database.global.datastore.id: 32542816-afce-f947-ab66-30e82835b7e
     mailnews.database.global.views.conversation.columns: {"threadCol":{"visible":true,"ordinal":"1"},"flaggedCol":{"visible":true,"ordinal":"21"},"attachmentCol":{"visible":fals…
     network.cookie.cookieBehavior: 3
     network.cookie.lifetimePolicy: 2
     network.cookie.prefsMigrated: true
     network.ftp.anonymous_password:
     network.predictor.cleaned-up: true
     places.database.lastMaintenance: 1435631093
     places.history.expiration.transient_current_max_pages: 104858
     places.history.expiration.transient_optimal_database_size: 167772160
     plugin.display_plugin_downloader_dialog: false
     plugin.importedState: true
     plugin.state.flash: 0
     print.always_print_silent: false
     print.macosx.pagesetup: PD94bWwgdmVyc2lvbj0iMS4wIiBlbmNvZGluZz0iVVRGLTgiPz4KPCFET0NUWVBFIHBsaXN0IFBVQkxJQyAiLS8vQXBwbGUgQ29tcHV0ZXIvL0RURCBQTElT…
     print.macosx.pagesetup-2: PD94bWwgdmVyc2lvbj0iMS4wIiBlbmNvZGluZz0iVVRGLTgiPz4KPCFET0NUWVBFIHBsaXN0IFBVQkxJQyAiLS8vQXBwbGUvL0RURCBQTElTVCAxLjAvL0VO…
     print.print_color: true
     print.print_command:
     print.print_evenpages: true
     print.print_file: false
     print.print_margin_bottom: 0.5
     print.print_margin_left: 0.5
     print.print_margin_right: 0.5
     print.print_margin_top: 0.5
     print.print_oddpages: true
     print.print_pagedelay: 500
     print.print_paper_size: 0
     print.print_reversed: false
     print.print_tofile:
     privacy.donottrackheader.enabled: true
     privacy.popups.first_popup: false
     privacy.popups.remove_blacklist: false
     security.disable_button.changePassword: false
     security.disable_button.openCertManager: false
     security.disable_button.openDeviceManager: false
     security.disable_button.resetPassword: false
     security.enable_java: false
     security.OCSP.signingCA:
     security.OCSP.URL:
     security.password_lifetime: 0

 Graphics

     Vendor ID: 0x1002
     Device ID: 0x9583
     WebGL Renderer: false
     GPU Accelerated Windows: 1/1 OpenGL

     AzureCanvasBackend: quartz
     AzureSkiaAccelerated: 0
     AzureFallbackCanvasBackend: none
     AzureContentBackend: quartz

 JavaScript

 Incremental GC: 1

 Accessibility

   Activated: 0
   Prevent Accessibility: 0

 Library Versions

     Expected minimum version
     Version in use

     NSPR
     4.10.8
     4.10.8

     NSS
     3.18.1 Basic ECC
     3.18.1 Basic ECC

     NSS Util
     3.18.1
     3.18.1

     NSS SSL
     3.18.1 Basic ECC
     3.18.1 Basic ECC

     NSS S/MIME
     3.18.1 Basic ECC
     3.18.1 Basic ECC

There is nothing in the Troubleshooting Information which stands out or would explain that behavior. So I don't really have anymore suggestions other than trying with a new profile.

The problem occurs on two different Macs running the same OS X and the same TB versions, same email accounts, same extensions (incl StartupMaster) and default themes.

I've now tried it with a new TB test profile on one of the Macs, with only one email acct defined and only the StartupMaster extension installed. The problem is unchanged whenever StartupMaster is enabled.

I'm in communication with the author of StartupMaster at the moment trying to get to the bottom of what's going on. I'll update this report when I know more.

Good news - upgrading both my Macs to the newly released TB 38.1 fixed the problem of the corrupted Thunderbird menu when using StartupMaster 1.6.1.

Now I can successfully use StartupMaster to work around the multiple password prompts issue (bug #177175) when launching Thunderbird.