I am often asked by my clients which Magento Extensions I can recommend for their Magento 1 or OpenMage shop. Thats why I created a (probably incomplete) list of awesome Magento or OpenMage Extensions that I regularly use in my Magento projects - next to my own Magento 1 extensions of course:

ExtensionCategoryMy comments
Mage Setup Administration Must-Have quickly adapt your Magento Store to German Market incl. tax, localizations, etc. (formerly called: German Setup)
TinyMCE HTML5 Compatibility Administration Must-Have fixes issues with HTML5 tags in WYSIWYG editor in Magento 1 backend
Attach Terms And Conditions Checkout Must-Have attaches agreement PDF documents (e.g. AGB) to any order email
AW Blog CMS nice, simple and effective blog extension (also used on my website)
AOE Scheduler Administration Must-Have huge improvement to the original Magento scheduler that offers a lot of actions in the backend for monitoring and crontrolling your Magento cronjobs
AutoShipping Checkout Must-Have automatically add shipping costs in cart so that the customer sees a relevant total amount value in the cart already
Change Attribute Sets Catalog provides the possibility to change attribute sets after product creation (this is not possible in Magento standard)
uDropship Catalog very huge (but also complex) extension for dropshipping capabilities
Recolize Recommendation Catalog Product Recommendation Engine
M2E Pro Catalog eBay/Amazon/Rakuten interface
SMTP Pro Extension Administration send your Magento emails through any email service
JS/CSS Timestamp Performance add last-modified timestamp to your JS and CSS merged files to enabled browser-based caching
HoneySpam Extension Security adds a little bit of bot protection to most of the forms, see more information in my blog post about captchas
Lesti FPC Extension Performance simple full page cache for Magento 1 / OpenMage

Another list of recommended Magento 1 extensions can also be found in this Github repository.

If you want to quickly start creating responsive templates I suggest to take a look at my blog article about Magento Mobile Themes.

I always suggest you take a look at the source code of each of the extensions you are installing in your shop. Remember that every Magento extension may do everything!

Looking forward to your comments on these!