Theme Check:

https://wordpress.org/plugins/theme-check/

Industry standard plugin for testing themes for most current best practices (!)

Plugin Performance Profiler:

https://wordpress.org/plugins/p3-profiler/

Allows to check how plugin performance affects the loading time of the WordPress website.

Google XML Sitemap Map generator:

https://wordpress.org/plugins/google-sitemap-generator/

Regenerate Thumbnails

http://www.viper007bond.com/wordpress-plugins/regenerate-thumbnails/

Useful when you need to do just what the plugin title describes

Custom Post Types:

https://wordpress.org/plugins/custom-post-type-ui/

A flexible tool for creating custom functionality for WordPress pages, by means of generating post templates that include content other than standard “blog posts” available in WordPress by default.

Advanced Custom Fields:

https://wordpress.org/plugins/advanced-custom-fields/

This plugin is necessary counterpart for the Custom Post Types plugin (they need each other in order to allow to build custom functionality into WordPress pages).

Types:

https://wordpress.org/plugins/types/

this is a “two-in-one” alternative to the two plugin above, preferred by some developers. Acceptable industry standard solution for custom functionality.

WooSlider:

http://www.woothemes.com/products/wooslider/

A popular commercial slider solution.

WordPress SEO by Yoast:

https://wordpress.org/plugins/wordpress-seo/

Offers the best support, has highest rating, and offers the highest flexibility among all WordPress on-page SEO plugins.

SEO Ultimate

https://wordpress.org/plugins/seo-ultimate/

(A possible alternative to WordPress SEO by Yoast)

Gravity Forms

http://www.gravityforms.com/

Considered top of the line premium forms plugin for WordPress – possibly the best. Many developers consider this a better alternative to Contact form 7 plugin.

Lightbox plus:

https://wordpress.org/plugins/lightbox-plus/

Automatically adds lightboxes in places where it makes sense.

Black Studio Tiny MCE WISIWIG Widget:

https://wordpress.org/plugins/black-studio-tinymce-widget/]

A great tool creating and easily editing custom widgets.

Display Widgets:

https://wordpress.org/plugins/display-widgets/

Easily control what widgets appear on what pages.

Google Analytics for WordPress from Yoast:

https://wordpress.org/plugins/google-analytics-for-wordpress/

Industry standard solution for implementing Google Analytics in WordPress.
Google Analytics Dashboard INSIDE WordPress Admin:

https://wordpress.org/plugins/google-analytics-dashboard-for-wp/

Displays all Google Analytics data inside WP-admin! Works well with Yoast GA plugin above.

Admin Menu Editor:

https://wordpress.org/plugins/admin-menu-editor/

Allows you to choose what is displayed on the admin panel.

iThemes Security / iThemes Security Pro:

https://wordpress.org/plugins/better-wp-security/

https://ithemes.com/security/

The industry standard solution for WordPress security.

Search and Replace:

https://wordpress.org/plugins/search-and-replace/

Purpose: find and replace all URLs to development site with the production (live) site URLs across all database tables, etc.)

(a better alternative to Velvet Blues)