The following table displays the permissions required by administrators for Apex classes:

Apex ClassEnabled
APTRUE
AbstractAgreementTermExActionControllerTRUE
AbstractApprovalEmailControllerTRUE
AbstractApprovalEngineProcessTermTRUE
AbstractApprovalsTestTRUE
AbstractQueryCallbackTRUE
AbstractSObjectApprovals2TRUE
AdhocApprovalGroupTRUE
AdhocApprovalProcessTRUE
AdhocApprovalProcessFactoryTRUE
AdhocApprovalTestSupportTRUE
AdhocApproverFactoryTRUE
AdhocProcessRuntimeControllerTestTRUE
AgreementApprovalEmailControllerTRUE
AgreementApprovalSummaryControllerTRUE
AgreementLIApprovalEmailControllerTestTRUE
AgreementLItemApprovalSummaryControllerTRUE
AgreementTEApprovalEmailControllerTestTRUE
AgreementTermExAddTestTRUE
AgreementTermExApprovalSummaryControllerTRUE
AgreementTermExApprovalsControllerTRUE
AgreementTermExCancelControllerTRUE
AgreementTermExEditControllerTRUE
AgreementTermExHistoryControllerTRUE
AgreementTermExPreviewControllerTRUE
AgreementTermExSubmitControllerTRUE
AgreementTermExWrapperTRUE
ApprovalActionTest2TRUE
ApprovalChildProcessTRUE
ApprovalConfigTestTRUE
ApprovalContextTRUE
ApprovalContextCancelTestTRUE
ApprovalContextPreviewControllerTRUE
ApprovalContextSubmitControllerTRUE
ApprovalCustomConfigTRUE
ApprovalDataTRUE
ApprovalDataTransferAgentTRUE
ApprovalEmailHandlerTRUE
ApprovalEngine2TRUE
ApprovalEngineConsolidationSupportTRUE
ApprovalEnginePrescanSupportTRUE
ApprovalEngineProcessAgreementTermExTRUE
ApprovalEngineProcessContextQueueableTRUE
ApprovalEngineProcessRequestTRUE
ApprovalEngineStepSupportTRUE
ApprovalEngineSupportTRUE
ApprovalEngineTestATESupportTRUE
ApprovalEngineTestAgmtSupportTRUE
ApprovalEngineTestOpptySupportTRUE
ApprovalHistoryControllerTestTRUE
ApprovalMatrixAssigneeSupportTRUE
ApprovalMatrixGenerateTestTRUE
ApprovalMatrixResolverTRUE
ApprovalMatrixViewControllerTRUE
ApprovalPolicySupportTRUE
ApprovalProcessDefnControllerTRUE
ApprovalProcessDefnListControllerTRUE
ApprovalProcessFactoryTRUE
ApprovalProcessFinalActionControllerTRUE
ApprovalProcessInitialActionControllerTRUE
ApprovalProcessInstanceTRUE
ApprovalProcessQryHelperTRUE
ApprovalProcessResolverTestTRUE
ApprovalProcessStepControllerTestTRUE
ApprovalProcessTest2TRUE
ApprovalRequestAttachmentControllerTRUE
ApprovalRequestEscalateControllerTRUE
ApprovalRequestQryHelperTRUE
ApprovalRequestReassignController2TRUE
ApprovalRequestSupportTRUE
ApprovalRequestsListTestTRUE
ApprovalRequiredCheckTestTRUE
ApprovalRuleAssigneeTRUE
ApprovalRuleControllerTRUE
ApprovalRuleCriteriaEditControllerTRUE
ApprovalRuleDimensionTRUE
ApprovalRuleDimensionControllerTestTRUE
ApprovalRuleDimensionSupportTRUE
ApprovalRuleEntryControllerTRUE
ApprovalRuleEntryFactoryTRUE
ApprovalRuleSupportTRUE
ApprovalStatusUpdateBatchJobTRUE
ApprovalSubprocessTRUE
ApprovalSummaryLaunchControllerTRUE
ApprovalSystemExceptionTRUE
ApprovalUserSupportTestTRUE
ApprovalsControllerTRUE
ApprovalsWebServiceTRUE
ApprovalsWebServiceTestTRUE
AssigneeCacheSupportTRUE
AsyncActionSupportTRUE
AsyncJobSupportTRUE
AttachmentWrapperTRUE
BackupAdminSupportTestTRUE
BackupApproverActionTestTRUE
BackupApproverSupportTRUE
BackupUserSearchControllerTRUE
BatchPreviewApprovalsTRUE
BulkActionRequestTRUE
ChildSOSupportTRUE
ClauseApprovalsWebServiceTestTRUE
ContextApprovalEmailControllerTestTRUE
CriteriaCacheTestTRUE
CurrencySupportTestTRUE
CustomAssigneeInfoTestTRUE
CustomClassTRUE
CustomSOSupportTRUE
DashboardSupportTestTRUE
ESAPISupportTRUE
EmailSupportTRUE
EmailTemplateFactoryTRUE
EmailTemplateTestTRUE
EscalationReminderEmailControllerTRUE
EscalationSupportTRUE
ExprSupportTRUE
FieldMetadataTRUE
FileSupportTestTRUE
FormulaFieldControllerTRUE
FormulaFieldFactoryTRUE
IApprovalEngineTRUE
InstanceUrlControllerTRUE
ManagedObjectTRUE
ManagedObjectTestTRUE
MyApprovalsControllerTRUE
MyApprovalsEmailControllerTRUE
MyApprovalsLaunchControllerTRUE
NextLevelApproverControllerTRUE
NullObjectTRUE
ObjectValueTRUE
OneOrMoreItemsFailedExceptionTRUE
OpportunityApprovalSummaryTestTRUE
OpportunityPartnersListTestTRUE
OpportunityProductsListTestTRUE
OpptyApprovalEmailControllerTestTRUE
PostInstallBatchApprovalRequestsTRUE
PostInstallBatchCopyProcessIdTRUE
PostInstallScriptTRUE
PreviewSubmitApprovalsControllerTRUE
PropertyTRUE
QuerySpecFactoryTRUE
ReassignUserSearchTestTRUE
ReminderInstanceSupportTRUE
ReminderJobTestTRUE
ReminderTestTRUE
RuntimeContextTRUE
SOQLSupportTRUE
SObjectApprovalContextParamTRUE
SObjectApprovals2SubmitControllerTRUE
SObjectApprovals2TestTRUE
SObjectApprovalsController2TRUE
SObjectApprovalsSupportTRUE
SObjectConstantsTRUE
SObjectMetadataTRUE
SObjectSupportTRUE
SearchFilterTRUE
SearchFilterControllerTestTRUE
SearchFilterDetailControllerTestTRUE
SubmissionCommentsTRUE
SystemUtilTRUE
TaskConstantsTRUE
TaskSupportTRUE
TermExApprovalCloneControllerTRUE
TermExApprovalControllerTRUE
TermExApprovalFactoryTRUE
TermExSupportTRUE
TransferInflightRequestsTRUE
UIAbstractSearchControllerTRUE
UIAssigneeControllerTRUE
UIDataTableControllerTRUE
UIDependsOnControllerTestTRUE
UIDisplayFieldNamesControllerTestTRUE
UIPageConfigTRUE
UIPageControllerBaseTRUE
UIPageSupportTRUE
UIReapprovalFilterControllerTestTRUE
UISearchEmailTemplateControllerTRUE
UISearchFilterControllerTestTRUE
UISearchRoleControllerTRUE
UISearchUserControllerTRUE

For information on setting Apex class permissions, refer to this Salesforce article.