We began randomly seeing a memory limit issue when attempting to view cart or checkout on the frontend, or create an order in the backend. After reviewing the logs, I disabled AvaTax and the error disappeared. No changes have been made recently and a mirror of the site on my development server did not experience the same issue. The memory limit on the production server is 512M and I ran the dev server down to 128M without any issue.
a:5:{i:0;s:85:"Allowed memory size of 536870912 bytes exhausted (tried to allocate 1768292353 bytes)";i:1;s:2579:"#0 /app/code/community/OnePica/AvaTax/lib/classes/DynamicSoapClient.class.php(27): SoapClient->SoapClient('file:///var/www...', Array)
#1 /app/code/community/OnePica/AvaTax/lib/classes/TaxServiceSoap.class.php(78): DynamicSoapClient->__construct('file:///var/www...', Array, Object(ATConfig))
#2 /app/code/community/OnePica/AvaTax/Model/Config.php(193): TaxServiceSoap->__construct('Magento')
#3 /app/code/community/OnePica/AvaTax/Model/Avatax/Abstract.php(54): OnePica_AvaTax_Model_Config->getTaxConnection()
#4 /app/code/community/OnePica/AvaTax/Model/Avatax/Estimate.php(181): OnePica_AvaTax_Model_Avatax_Abstract->_send(NULL)
#5 /app/code/community/OnePica/AvaTax/Model/Avatax/Estimate.php(109): OnePica_AvaTax_Model_Avatax_Estimate->_getRates(Object(Mage_Sales_Model_Quote_Item))
#6 /app/code/community/OnePica/AvaTax/Model/Sales/Quote/Address/Total/Tax.php(63): OnePica_AvaTax_Model_Avatax_Estimate->getItemTax(Object(Mage_Sales_Model_Quote_Item))
#7 /app/code/core/Mage/Sales/Model/Quote/Address.php(1013): OnePica_AvaTax_Model_Sales_Quote_Address_Total_Tax->collect(Object(OnePica_AvaTax_Model_Sales_Quote_Address))
#8 /app/code/core/Mage/Sales/Model/Quote.php(1331): Mage_Sales_Model_Quote_Address->collectTotals()
#9 /app/code/core/Mage/Checkout/Model/Cart.php(458): Mage_Sales_Model_Quote->collectTotals()
#10 /app/code/core/Mage/Checkout/controllers/CartController.php(144): Mage_Checkout_Model_Cart->save()
#11 /app/code/core/Mage/Core/Controller/Varien/Action.php(418): Mage_Checkout_CartController->indexAction()
#12 /app/code/local/Mage/Core/Controller/Varien/Router/Standard.php(254): Mage_Core_Controller_Varien_Action->dispatch('index')
#13 /app/code/core/Mage/Core/Controller/Varien/Front.php(172): Mage_Core_Controller_Varien_Router_Standard->match(Object(Mage_Core_Controller_Request_Http))
#14 /app/code/core/Mage/Core/Model/App.php(365): Mage_Core_Controller_Varien_Front->dispatch()
#15 /app/Mage.php(684): Mage_Core_Model_App->run(Array)
#16 /index.php(88): Mage::run('store_englis...', 'store')
#17 {main}";s:3:"url";s:15:"/checkout/cart/";s:11:"script_name";s:10:"/index.php";s:4:"skin";s:16:"store_english";}