Quick Page/Post Redirect Plugin: Allows you to create redirects from a Request URL (existing or non-existing page) to a new destination URL, all without touching .htaccess or updating cPanel Redirects.
Wordfence Security: The best WordPress security suite I've found for monitoring for attacks, implementing firewalls, country blocking, brute force login attempts, monitoring that themes and plugins are up-to-date, etc.
s2Member Pro: Membership management plugin. Limits access to content based on the membership status, membership level, and assigned custom capabilities.
Duplicator: A great plugin that allows you to take a full backup of a WordPress installation, move a few files over to a new server and it will completely recreate the site on to the new server, including updating URL's if they have changed, creating new Admin users, etc.
Global Content Blocks: [It looks like this plugin might be no longer in development and it's been removed from the WordPress Plugin Directory.] If you need to re-use content in multiple areas of the site, but want to edit them all in just one location, Global Content Blocks is a great solution. It allows creation of HTML, Code or PHP types of content blocks, so you can execute PHP scripts within your blocks. Possible replacements would be Infuse or Reusable Text Blocks (both untested.) On old sites I have used WP-Components but it hasn't been updated since 2009.
WooCommerce: An awesome shopping cart solution that has a lot of plugins such as Stripe for payment processing, and direct ties to USPS and UPS for shipping calculation.
Imsanity: Finds large images on the site and can reduce their size, as well as monitoring for new files being uploaded and automatically scale them down.
Gravity Forms: A great form processor that can tie in with many other systems, such as Stripe for credit card payment processing.
Gravity Forms: Force SSL: If you only want to secure your forms instead of the whole site (why!?) this plugin will do the trick.
Really Simple SSL: Redirect all non-SSL pages back to their SSL versions. Getting mixed content errors try this or this (or use the Firefox Web Console).
Flexible Posts Widget: Allows you to control how posts are displayed (pages, widgets, etc.) Used when I want to display one large most recent post, and then a collection of smaller older posts.
Child Theme Configurator: It's so easy to create a Child Theme manually, but once in a while some of the settings don't copy over and the child theme looks totally different (menus are missing, widgets are wrong, etc.) By using this, when you activate the Child Theme I've typically had it look exactly the same as the parent theme.
WP Easy Uploader: On occasion I need my clients to be able to upload files to places other than the Media Library and want them to be able to do that through the regular WordPress Dashboard. This plugin allows them to assign a custom path and upload the files directly. If the directory doesn't exist it will attempt to create it on it's own. Hasn't been updated since 2009, there are likely alternatives that are better and more secure.
WP Sync DB: When working on local development sites, or remote dev sites and you need to sync data from the live site back to them or vice-versa, this is a tool that will get the job done for you. It will also sync your Media Library content between the sites if desired as well.
Postman SMTP Mailer/Email Log: For servers that can't send mail directly and need to route mail through an existing Google Apps account, Postman SMTP is a good plugin for the job. (The problem was actually caused by this WordPress 4.6 bug.)
Check Email: A plugin that simply allows you to test if WordPress can send emails properly. [I now typically just use the email tool inside Wordfence to test sending emails.]
Media Tools: When you use the built in WordPress Import/Export tools, it will import your posts or pages, but it does not (typically) actually copy over the images in to the Media Library, it just leaves them referencing the files on the old server. For a small site, you just copy over the /uploads/ directory and do a find and replace on the URL's to get them updated to the new site. But what if you want to replace all of those old references with local references and actually add the media to your Media Library (for example, only published posts, or posts within a specific date range and not everything?) Well, Media Tools is the answer. It'll go through all of your posts/pages, etc, find any references to external images, download a copy of it, add it to your Media Library, and update the URL in the post to point to the local version.
Velvet Blues Update URLs: This was my favorite tool to find and replace URL's in the database when transitioning from a development site URL to the live site URL. I found on some sites (ProPhoto) that it didn't go through all of the database tables/fields. So when Velvet Blues doesn't work I have used Search Replace DB, which isn't a WordPress plugin but certainly gets the job done. Another good option is Better Search Replace. And lately I've had good luck with Search & Replace and liked that it has a Dry Run feature.