The following table displays the permissions required by administrators for users:

Apex ClassEnabled
APTRUE
AbstractAdhocApprovalEngineTRUE
AbstractAgreementTermExActionControllerTRUE
AbstractApprovalContextActionControllerTRUE
AbstractApprovalEmailControllerTRUE
AbstractApprovalEngineTRUE
AbstractApprovalEngineProcessTermTRUE
AbstractApprovalEngineTestTRUE
AbstractApprovalsTestTRUE
AbstractEngineTRUE
AbstractQueryCallbackTRUE
AbstractSObjectApprovalsTRUE
AbstractSObjectApprovals2TRUE
AdhocApprovalEngineTRUE
AdhocApprovalGroupTRUE
AdhocApprovalGroupFactoryTRUE
AdhocApprovalProcessTRUE
AdhocApprovalProcessDTOTRUE
AdhocApprovalProcessFactoryTRUE
AdhocApprovalTestTRUE
AdhocApprovalTestSupportTRUE
AdhocApproverTRUE
AdhocApproverFactoryTRUE
AdhocProcessRuntimeControllerTRUE
AdhocProcessRuntimeControllerTestTRUE
AdhocProcessRuntimeSupportTRUE
AgreementApprovalEmailControllerTRUE
AgreementApprovalEmailControllerTestTRUE
AgreementApprovalSummaryControllerTRUE
AgreementApprovalSummaryControllerTestTRUE
AgreementLIApprovalEmailControllerTestTRUE
AgreementLIApprovalSummaryControllerTestTRUE
AgreementLItemApprovalSummaryControllerTRUE
AgreementLineItemApprovalEmailControllerTRUE
AgreementTEApprovalEmailControllerTestTRUE
AgreementTermExAddControllerTRUE
AgreementTermExAddTestTRUE
AgreementTermExApprovalEmailControllerTRUE
AgreementTermExApprovalSummaryControllerTRUE
AgreementTermExApprovalSummaryTestTRUE
AgreementTermExApprovalsControllerTRUE
AgreementTermExApprovalsTestTRUE
AgreementTermExCancelControllerTRUE
AgreementTermExCancelTestTRUE
AgreementTermExEditControllerTRUE
AgreementTermExEditTestTRUE
AgreementTermExHistoryControllerTRUE
AgreementTermExHistoryTestTRUE
AgreementTermExPreviewControllerTRUE
AgreementTermExPreviewTestTRUE
AgreementTermExSubmitControllerTRUE
AgreementTermExSubmitTestTRUE
AgreementTermExWrapperTRUE
ApprovalActionController2TRUE
ApprovalActionTest2TRUE
ApprovalAssigneeResolverTRUE
ApprovalChildProcessTRUE
ApprovalConfigTRUE
ApprovalConfigTestTRUE
ApprovalConstantsTRUE
ApprovalContextTRUE
ApprovalContextCancelControllerTRUE
ApprovalContextCancelTestTRUE
ApprovalContextFactoryTRUE
ApprovalContextPreviewControllerTRUE
ApprovalContextPreviewTestTRUE
ApprovalContextSubmitControllerTRUE
ApprovalContextSubmitTestTRUE
ApprovalCustomConfigTRUE
ApprovalCustomConfigTestTRUE
ApprovalDataTRUE
ApprovalDataTestTRUE
ApprovalDataTransferAgentTRUE
ApprovalDeleteSupportTRUE
ApprovalEmailHandlerTRUE
ApprovalEmailHandlerTestTRUE
ApprovalEngine2TRUE
ApprovalEngineBatchActionHandlerTRUE
ApprovalEngineConsolidationSupportTRUE
ApprovalEngineFactoryTRUE
ApprovalEnginePrescanSupportTRUE
ApprovalEnginePreviewHelperTRUE
ApprovalEngineProcessAgreementTermExTRUE
ApprovalEngineProcessContextObjectTRUE
ApprovalEngineProcessContextQueueableTRUE
ApprovalEngineProcessDealTermTRUE
ApprovalEngineProcessRequestTRUE
ApprovalEngineProcessRequestQueueableTRUE
ApprovalEngineStepSupportTRUE
ApprovalEngineSubmitHelperTRUE
ApprovalEngineSupportTRUE
ApprovalEngineTestATETRUE
ApprovalEngineTestATESupportTRUE
ApprovalEngineTestAgmtTRUE
ApprovalEngineTestAgmtSupportTRUE
ApprovalEngineTestOpptyTRUE
ApprovalEngineTestOpptySupportTRUE
ApprovalHistoryControllerTRUE
ApprovalHistoryControllerTestTRUE
ApprovalHistorySupportTRUE
ApprovalMatrixAssigneeSupportTRUE
ApprovalMatrixGenerateControllerTRUE
ApprovalMatrixGenerateTestTRUE
ApprovalMatrixGeneratorTRUE
ApprovalMatrixResolverTRUE
ApprovalMatrixSupportTRUE
ApprovalMatrixViewControllerTRUE
ApprovalMatrixViewTestTRUE
ApprovalPolicySupportTRUE
ApprovalProcess2TRUE
ApprovalProcessDefnControllerTRUE
ApprovalProcessDefnControllerTestTRUE
ApprovalProcessDefnListControllerTRUE
ApprovalProcessDefnListTestTRUE
ApprovalProcessFactoryTRUE
ApprovalProcessFactoryTestTRUE
ApprovalProcessFinalActionControllerTRUE
ApprovalProcessFinalActionTestTRUE
ApprovalProcessInitialActionControllerTRUE
ApprovalProcessInitialActionTestTRUE
ApprovalProcessInstanceTRUE
ApprovalProcessInstanceFactoryTRUE
ApprovalProcessQryHelperTRUE
ApprovalProcessResolverTRUE
ApprovalProcessResolverTestTRUE
ApprovalProcessStepControllerTRUE
ApprovalProcessStepControllerTestTRUE
ApprovalProcessSupportTRUE
ApprovalProcessTest2TRUE
ApprovalQryHelperTRUE
ApprovalRequestAttachmentControllerTRUE
ApprovalRequestAttachmentTestTRUE
ApprovalRequestEscalateControllerTRUE
ApprovalRequestEscalateTestTRUE
ApprovalRequestQryHelperTRUE
ApprovalRequestQryHelperTestTRUE
ApprovalRequestReassignController2TRUE
ApprovalRequestReassignTestTRUE
ApprovalRequestSupportTRUE
ApprovalRequestsListControllerTRUE
ApprovalRequestsListTestTRUE
ApprovalRequiredCheckTRUE
ApprovalRequiredCheckTestTRUE
ApprovalRule2TRUE
ApprovalRuleAssigneeTRUE
ApprovalRuleAssigneeFactoryTRUE
ApprovalRuleControllerTRUE
ApprovalRuleControllerTestTRUE
ApprovalRuleCriteriaEditControllerTRUE
ApprovalRuleCriteriaEditControllerTestTRUE
ApprovalRuleDimensionTRUE
ApprovalRuleDimensionControllerTRUE
ApprovalRuleDimensionControllerTestTRUE
ApprovalRuleDimensionFactoryTRUE
ApprovalRuleDimensionSupportTRUE
ApprovalRuleEntryTRUE
ApprovalRuleEntryControllerTRUE
ApprovalRuleEntryControllerTestTRUE
ApprovalRuleEntryFactoryTRUE
ApprovalRuleFactoryTRUE
ApprovalRuleSupportTRUE
ApprovalRuleTestTRUE
ApprovalStatusUpdateBatchJobTRUE
ApprovalStatusUpdateBatchJobTestTRUE
ApprovalSubprocessTRUE
ApprovalSubstepTRUE
ApprovalSummaryLaunchControllerTRUE
ApprovalSummaryLaunchTestTRUE
ApprovalSystemExceptionTRUE
ApprovalUserSupportTRUE
ApprovalUserSupportTestTRUE
ApprovalUtil2TRUE
ApprovalsControllerTRUE
ApprovalsControllerTestTRUE
ApprovalsWebServiceTRUE
ApprovalsWebServiceSupportTRUE
ApprovalsWebServiceTestTRUE
AssigneeTRUE
AssigneeCacheSupportTRUE
AsyncActionInfoEmailControllerTRUE
AsyncActionSupportTRUE
AsyncActionSupportTestTRUE
AsyncJobSupportTRUE
AttachmentSupportTRUE
AttachmentWrapperTRUE
BackupAdminSupportTRUE
BackupAdminSupportTestTRUE
BackupApproverActionControllerTRUE
BackupApproverActionTestTRUE
BackupApproverControllerTRUE
BackupApproverSupportTRUE
BackupApproverTestTRUE
BackupUserSearchControllerTRUE
BackupUserSearchTestTRUE
BatchPreviewApprovalsTRUE
BatchSubmitForApprovalsTRUE
BulkActionRequestTRUE
BulkCtxObjectParamTRUE
ChildSOSupportTRUE
ClauseApprovalsWebServiceTRUE
ClauseApprovalsWebServiceTestTRUE
ContextApprovalEmailControllerTRUE
ContextApprovalEmailControllerTestTRUE
CriteriaCacheTRUE
CriteriaCacheTestTRUE
CurrencySupportTRUE
CurrencySupportTestTRUE
CustomAssigneeInfoTRUE
CustomAssigneeInfoTestTRUE
CustomAssigneeSupportTRUE
CustomClassTRUE
CustomClassLoaderTRUE
CustomSOSupportTRUE
DashboardSupportTRUE
DashboardSupportTestTRUE
DefaultQueryCallbackTRUE
ESAPISupportTRUE
EmailRecipientTRUE
EmailSupportTRUE
EmailTemplateConstantsTRUE
EmailTemplateFactoryTRUE
EmailTemplateSupportTRUE
EmailTemplateTestTRUE
EscalationAssigneeTRUE
EscalationReminderEmailControllerTRUE
EscalationReminderEmailControllerTestTRUE
EscalationSupportTRUE
EscalationWorkflowTestTRUE
ExprSupportTRUE
ExprTestTRUE
FieldMetadataTRUE
FileSupportTRUE
FileSupportTestTRUE
FormulaFieldTRUE
FormulaFieldControllerTRUE
FormulaFieldControllerTestTRUE
FormulaFieldFactoryTRUE
FormulaFieldSupportTRUE
IApprovalEngineTRUE
InboundAttachmentHandlerTestTRUE
InstanceUrlControllerTRUE
LookupUserSupportTRUE
ManagedObjectTRUE
ManagedObjectConstantsTRUE
ManagedObjectTestTRUE
MiscClassTestTRUE
MyApprovalsControllerTRUE
MyApprovalsControllerTestTRUE
MyApprovalsEmailControllerTRUE
MyApprovalsEmailControllerTestTRUE
MyApprovalsLaunchControllerTRUE
MyApprovalsLaunchControllerTestTRUE
NextLevelApproverControllerTRUE
NextLevelApproverTestTRUE
NullObjectTRUE
ObjectCacheTRUE
ObjectValueTRUE
ObjectValueTestTRUE
OneOrMoreItemsFailedExceptionTRUE
OpportunityApprovalSummaryControllerTRUE
OpportunityApprovalSummaryTestTRUE
OpportunityPartnersListControllerTRUE
OpportunityPartnersListTestTRUE
OpportunityProductsListControllerTRUE
OpportunityProductsListTestTRUE
OpptyApprovalEmailControllerTRUE
OpptyApprovalEmailControllerTestTRUE
PostInstallBatchApprovalProcessesTRUE
PostInstallBatchApprovalRequestsTRUE
PostInstallBatchBackupApproversTRUE
PostInstallBatchCopyProcessIdTRUE
PostInstallBatchUpdDelegateApproverNamesTRUE
PostInstallScriptTRUE
PostInstallScriptTestTRUE
PreviewSubmitApprovalsControllerTRUE
PreviewSubmitApprovalsTestTRUE
PropertyTRUE
QuerySpecTRUE
QuerySpecFactoryTRUE
ReassignUserSearchControllerTRUE
ReassignUserSearchTestTRUE
RelatedUserSupportTRUE
ReminderInstanceSupportTRUE
ReminderJobTRUE
ReminderJobTestTRUE
ReminderSupportTRUE
ReminderTestTRUE
ReminderTestSupportTRUE
RuntimeContextTRUE
SOQLConstantsTRUE
SOQLSupportTRUE
SOQLTestTRUE
SObjectApprovalContextParamTRUE
SObjectApprovalContextParam2TRUE
SObjectApprovals2SubmitControllerTRUE
SObjectApprovals2SubmitTestTRUE
SObjectApprovals2TestTRUE
SObjectApprovalsControllerTRUE
SObjectApprovalsController2TRUE
SObjectApprovalsSampleTRUE
SObjectApprovalsSupportTRUE
SObjectApprovalsTestTRUE
SObjectConstantsTRUE
SObjectDescribeInfoTRUE
SObjectMetadataTRUE
SObjectMetadataTestTRUE
SObjectSupportTRUE
SObjectTestTRUE
SearchFilterTRUE
SearchFilterControllerTRUE
SearchFilterControllerTestTRUE
SearchFilterDetailControllerTRUE
SearchFilterDetailControllerTestTRUE
StepRuleTRUE
SubmissionCommentsTRUE
SubmissionCommentsSupportTRUE
SystemUtilTRUE
SystemUtilTestTRUE
TaskConstantsTRUE
TaskFactoryTRUE
TaskSupportTRUE
TermExApprovalTRUE
TermExApprovalCloneControllerTRUE
TermExApprovalCloneTestTRUE
TermExApprovalControllerTRUE
TermExApprovalControllerTestTRUE
TermExApprovalFactoryTRUE
TermExApprovalFactoryTestTRUE
TermExSupportTRUE
TestSupportTRUE
TransferInflightRequestsTRUE
TransferInflightRequestsQueueableTRUE
UIAbstractSearchControllerTRUE
UIAbstractSelectItemsControllerTRUE
UIAssigneeControllerTRUE
UIAssigneeControllerTestTRUE
UIDataTableControllerTRUE
UIDependsOnControllerTRUE
UIDependsOnControllerTestTRUE
UIDisplayFieldNamesControllerTRUE
UIDisplayFieldNamesControllerTestTRUE
UIEscalationAssigneeControllerTRUE
UIPageConfigTRUE
UIPageConfigTestTRUE
UIPageControllerBaseTRUE
UIPageControllerBaseTestTRUE
UIPageSupportTRUE
UIReapprovalFilterControllerTRUE
UIReapprovalFilterControllerTestTRUE
UISearchControllerTestTRUE
UISearchEmailTemplateControllerTRUE
UISearchFilterControllerTRUE
UISearchFilterControllerTestTRUE
UISearchQueueControllerTRUE
UISearchRoleControllerTRUE
UISearchRuleControllerTRUE
UISearchUserControllerTRUE
UserContextTRUE

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