aura:if breaks again.

demo1.app

<aura:application >
    <aura:attribute name="value1" type="String" access="private" />
    <aura:attribute name="value2" type="String" access="private" />
    <lightning:input value="{!v.value1}" label="Value 1" placeholder="Enter 'a'" />
    <aura:if isTrue="{!v.value1 eq 'a'}">
        <lightning:input value="{!v.value2}" label="Value 2" placeholder="Enter 'b'" />
        <aura:if isTrue="{!v.value2 eq 'b'}">
            <lightning:input label="Value 3" value="{!join(' ',v.value1,v.value2)}" />
        </aura:if>
    </aura:if>
    <!--
        To make the error happen:
         type in 'a' in the first box,
         'b' in the second box,
         then change the value of box 1 twice.

      Access Check Failed! AttributeSet.get(): attribute 'value2' of component 'markup://c:demo1 {1:0}'
      is not visible to 'markup://aura:if {6:0}'.
      Failing descriptor: {aura:if}

  -->
</aura:application>

demo1fixed.app

<aura:application extends="force:slds">
    <aura:attribute name="value1" type="String" access="private" />
    <aura:attribute name="value2" type="String" access="private" />
    <lightning:input value="{!v.value1}" label="Value 1" placeholder="Enter 'a'" />
    <div class="{!v.value1 eq 'a'?'':'slds-hide'}">
        <lightning:input value="{!v.value2}" label="Value 2" placeholder="Enter 'b'" />
        <div class="{!v.value2 eq 'b'?'':'slds-hide'}">
            <lightning:input label="Value 3" value="{!join(' ',v.value1,v.value2)}" />
        </div>
    </div>
	<!-- 
	Filed as W-5722782.

	This workaround will not produce the error.
  
  If not using SLDS, write your own style that's basically "display: none".
	-->
</aura:application>

The basic bug is that aura:if has a problem when it has to rerender a lightning component (any component in the lightning namespace, but not aura or ui components, etc). This is a side effect of LWC, essentially. It should only occur with nested aura:if as far as I can tell.

Either use slds-hide or custom CSS, do not use lightning:input or other components, do not nest aura:if, or do not make your attributes private.